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Solution - Derivative

\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) - \frac{6 a c r \cos (\frac{3}{x^{2}})}{x^{3}}

\frac{d}{dx}[a]\times rc\times \sin{\left(\frac{3}{x^{2}} \right)}+a\times \frac{d}{dx}[r]\times c\times \sin{\left(\frac{3}{x^{2}} \right)}+ar\times \frac{d}{dx}[c]\times \sin{\left(\frac{3}{x^{2}} \right)}- \frac{6 a c r \cos{\left(\frac{3}{x^{2}} \right)}}{x^{3}}

See steps

Step-by-step explanation

1. Solve derivative

19 additional steps

Expanding the derivative for multiplication.

$\frac{d}{d x} [a r c \times \sin (\frac{3}{x^{2}})] = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]$

Expanding the derivative for multiplication.

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [a r c \times \sin (\frac{3}{x^{2}})] = \frac{d}{d x} [a \times (r c \times \sin (\frac{3}{x^{2}}))]$

Applying the product rule of derivatives.

$\frac{d}{d x} [a \times (r c \times \sin (\frac{3}{x^{2}}))] = \frac{d}{d x} [a] \times (r c \times \sin (\frac{3}{x^{2}})) + a \times \frac{d}{d x} [r c \times \sin (\frac{3}{x^{2}})]$

Expanding the derivative for multiplication.

$\frac{d}{d x} [a] \times (r c \times \sin (\frac{3}{x^{2}})) + a \times \frac{d}{d x} [r c \times \sin (\frac{3}{x^{2}})] = \frac{d}{d x} [a] \times (r c \times \sin (\frac{3}{x^{2}})) + a (\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [r c \times \sin (\frac{3}{x^{2}})] = \frac{d}{d x} [r \times (c \times \sin (\frac{3}{x^{2}}))]$

Applying the product rule of derivatives.

$\frac{d}{d x} [r \times (c \times \sin (\frac{3}{x^{2}}))] = \frac{d}{d x} [r] \times (c \times \sin (\frac{3}{x^{2}})) + r \times \frac{d}{d x} [c \times \sin (\frac{3}{x^{2}})]$

Expanding the derivative for multiplication.

Applying the product rule of derivatives.

$\frac{d}{d x} [c \times \sin (\frac{3}{x^{2}})] = \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [r] \times (c \times \sin (\frac{3}{x^{2}})) + r (\frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]) = \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + r (\frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])$

Multiplying a number by a sum or difference of two numbers can be done by multiplying each number individually and then adding or subtracting the results.

$\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + r (\frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]) = \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + (r \times (\frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}})) + r \times (c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + (r \times (\frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}})) + r \times (c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])) = \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r \times (c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r \times (c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])) = \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])$

Addition can be grouped differently, but the result remains the same.

$\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]) = \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [a] \times (r c \times \sin (\frac{3}{x^{2}})) + a (\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a (\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])$

Multiplying a number by a sum or difference of two numbers can be done by multiplying each number individually and then adding or subtracting the results.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a (\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times (\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}})) + a \times (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}})) + a \times (r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times (\frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}})) + a \times (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}})) + a \times (r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a \times (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}})) + a \times (r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a \times (r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}})) + a \times (r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a \times (r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a \times (r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})])$

Addition can be grouped differently, but the result remains the same.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + (a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})]$

2 additional steps

Computing the derivative of a sine function using the chain rule.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times \frac{d}{d x} [\sin (\frac{3}{x^{2}})] = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{d}{d x} [\frac{3}{x^{2}}])$

Decomposing the function for the chain rule.

$\frac{d}{d x} [\sin (\frac{3}{x^{2}})] = \frac{d}{d x} [\sin (x)] \times \frac{d}{d x} [\frac{3}{x^{2}}]$

Computing the derivative of a sine function.

$\frac{d}{d x} [\sin (x)] \times \frac{d}{d x} [\frac{3}{x^{2}}] = \cos (x) \times \frac{d}{d x} [\frac{3}{x^{2}}]$

Substituting the variable back into the function.

$\cos (x) \times \frac{d}{d x} [\frac{3}{x^{2}}] = \cos (\frac{3}{x^{2}}) \times \frac{d}{d x} [\frac{3}{x^{2}}]$

Computing the derivative of a fraction.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{d}{d x} [\frac{3}{x^{2}}]) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{\frac{d}{d x} [3] \times x^{2} - 3 \times \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}})$

The derivative of a constant value is always zero.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{\frac{d}{d x} [3] \times x^{2} - 3 \times \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}})$

Computing the derivative of x raised to the power of n.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times (2 x^{2 - 1})}{{(x^{2})}^{2}})$

Subtracting one from a number.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times (2 x^{2 - 1})}{{(x^{2})}^{2}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times (2 x^{1})}{{(x^{2})}^{2}})$

Any number raised to the power of one equals the number itself.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times (2 x^{1})}{{(x^{2})}^{2}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times (2 x)}{{(x^{2})}^{2}})$

Multiplying a number by zero always results in zero.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 x^{2} - 3 \times (2 x)}{{(x^{2})}^{2}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 - 3 \times (2 x)}{{(x^{2})}^{2}})$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 - 3 \times (2 x)}{{(x^{2})}^{2}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 - 3 \times (2 x)}{x^{4}})$

Adding zero to a number, which does not change its value.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{0 - 3 \times (2 x)}{x^{4}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{- 3 \times (2 x)}{x^{4}})$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{- 3 \times (2 x)}{x^{4}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{- 6 x}{x^{4}})$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times \frac{- 6 x}{x^{4}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times (- \frac{6}{x^{3}}))$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (\cos (\frac{3}{x^{2}}) \times (- \frac{6}{x^{3}})) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (- \frac{6 \cos (\frac{3}{x^{2}})}{x^{3}})$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) + a r c \times (- \frac{6 \cos (\frac{3}{x^{2}})}{x^{3}}) = \frac{d}{d x} [a] \times r c \times \sin (\frac{3}{x^{2}}) + a \times \frac{d}{d x} [r] \times c \times \sin (\frac{3}{x^{2}}) + a r \times \frac{d}{d x} [c] \times \sin (\frac{3}{x^{2}}) - \frac{6 a c r \cos (\frac{3}{x^{2}})}{x^{3}}$

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Why learn this

Ever wondered how to predict the future? Derivatives are your crystal ball!

Picture this: You're a surfer trying to catch the biggest wave. How do you know when it's coming? Derivatives can tell you when it's at its highest point!

Rocket Science: Planning to launch a rocket to Mars? Derivatives tell us the optimal fuel burn rate to minimize fuel consumption and maximize distance!

Stock Market: Trading in the stock market? Derivatives can indicate the rate at which stock prices are changing, helping predict the best time to buy or sell.

Animation: Love animated movies? Artists use derivatives to smoothly change the motion and expressions of characters, making them feel more lifelike.

Engineering: Designing a bridge or a skyscraper? Derivatives help determine the rates of stress and strain changes in materials, ensuring the safety of your structures.

In short, derivatives are like a secret code to understanding change and making predictions in real life. So let's crack this code together and become masters of our futures!

Terms and topics

Derivative

Solution - Derivative

Step-by-step explanation

1. Solve derivative

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