คำตอบ - Derivative

- \frac{15 x^{2} \cos (x)}{\sin^{2} (5 x^{3})}

- \frac{15 x^{2} \cos{\left(x \right)}}{\sin^{2}{\left(5 x^{3} \right)}}

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6 additional steps

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

$\frac{d}{d x} [\csc (5 x^{3})] = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times \frac{d}{d x} [5 x^{3}]$

Decomposing the function for the chain rule.

$\frac{d}{d x} [\csc (5 x^{3})] = \frac{d}{d x} [\csc (x)] \times \frac{d}{d x} [5 x^{3}]$

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

$\frac{d}{d x} [\csc (5 x^{3})] = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times \frac{d}{d x} [5 x^{3}]$

The cosecant of an angle is equal to one divided by the sine of the angle.

$\frac{d}{d x} [\csc (x)] = \frac{d}{d x} [\frac{1}{\sin (x)}]$

Computing the derivative of a reciprocal.

$\frac{d}{d x} [\frac{1}{\sin (x)}] = - \frac{1}{\sin {(x)}^{2}} \times \frac{d}{d x} [\sin (x)]$

Computing the derivative of a sine function.

$- \frac{1}{\sin {(x)}^{2}} \times \frac{d}{d x} [\sin (x)] = - \frac{1}{\sin {(x)}^{2}} \times \cos (x)$

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

$(- \frac{1}{\sin {(x)}^{2}} \times \cos (x)) \times \frac{d}{d x} [5 x^{3}] = - \frac{1}{\sin {(x)}^{2}} \times \cos (x) \times \frac{d}{d x} [5 x^{3}]$

Substituting the variable back into the function.

$- \frac{1}{\sin {(x)}^{2}} \times \cos (x) \times \frac{d}{d x} [5 x^{3}] = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times \frac{d}{d x} [5 x^{3}]$

Applying the product rule of derivatives.

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times \frac{d}{d x} [5 x^{3}] = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (\frac{d}{d x} [5] \times x^{3} + 5 \times \frac{d}{d x} [x^{3}])$

The derivative of a constant value is always zero.

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (\frac{d}{d x} [5] \times x^{3} + 5 \times \frac{d}{d x} [x^{3}]) = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (0 x^{3} + 5 \times \frac{d}{d x} [x^{3}])$

Multiplying a number by zero always results in zero.

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (0 x^{3} + 5 \times \frac{d}{d x} [x^{3}]) = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (0 + 5 \times \frac{d}{d x} [x^{3}])$

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

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (0 + 5 \times \frac{d}{d x} [x^{3}]) = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (5 \times \frac{d}{d x} [x^{3}])$

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

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (5 \times \frac{d}{d x} [x^{3}]) = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (5 \times (3 x^{3 - 1}))$

Subtracting one from a number.

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (5 \times (3 x^{3 - 1})) = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (5 \times (3 x^{2}))$

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

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (5 \times (3 x^{2})) = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times ((5 \times 3) \times x^{2})$

Multiplying two integers together.

$- \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times ((5 \times 3) \times x^{2}) = - \frac{1}{\sin {(5 x^{3})}^{2}} \times \cos (x) \times (15 x^{2})$