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Solution - Derivative

x^{2} y^{3} + x \times (y^{2} \times (\frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + 2 x^{2} y^{3} + x y^{2} x^{2} \times \frac{d}{d x} [y]

x^{2} y^{3}+x\times (y^{2}\times (\frac{2}{y}\times \frac{d}{dx}[y]))\times x^{2}y+2 x^{2} y^{3}+xy^{2}x^{2}\times \frac{d}{dx}[y]

See steps

Step-by-step explanation

1. Solve derivative

19 additional steps

Expanding the derivative for multiplication.

$\frac{d}{d x} [x y^{2} x^{2} y] = \frac{d}{d x} [x] \times y^{2} x^{2} y + x \times \frac{d}{d x} [y^{2}] \times x^{2} y + x y^{2} \times \frac{d}{d x} [x^{2}] \times y + x y^{2} x^{2} \times \frac{d}{d x} [y]$

Expanding the derivative for multiplication.

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

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

Applying the product rule of derivatives.

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

Expanding the derivative for multiplication.

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

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

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

Applying the product rule of derivatives.

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

Expanding the derivative for multiplication.

Applying the product rule of derivatives.

$\frac{d}{d x} [x^{2} y] = \frac{d}{d x} [x^{2}] \times y + x^{2} \times \frac{d}{d x} [y]$

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

The derivative of a variable with respect to itself is always equal to one.

$\frac{d}{d x} [x] \times y^{2} x^{2} y + x \times \frac{d}{d x} [y^{2}] \times x^{2} y + x y^{2} \times \frac{d}{d x} [x^{2}] \times y + x y^{2} x^{2} \times \frac{d}{d x} [y] = 1 y^{2} x^{2} y + x \times \frac{d}{d x} [y^{2}] \times x^{2} y + x y^{2} \times \frac{d}{d x} [x^{2}] \times y + x y^{2} x^{2} \times \frac{d}{d x} [y]$

Computing the derivative of a power function.

$1 y^{2} x^{2} y + x \times \frac{d}{d x} [y^{2}] \times x^{2} y + x y^{2} \times \frac{d}{d x} [x^{2}] \times y + x y^{2} x^{2} \times \frac{d}{d x} [y] = 1 y^{2} x^{2} y + x \times (y^{2} (\frac{d}{d x} [2] \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times \frac{d}{d x} [x^{2}] \times y + x y^{2} x^{2} \times \frac{d}{d x} [y]$

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

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

Subtracting one from a number.

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

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

$1 y^{2} x^{2} y + x \times (y^{2} (\frac{d}{d x} [2] \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x^{1}) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y] = 1 y^{2} x^{2} y + x \times (y^{2} (\frac{d}{d x} [2] \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y]$

Simplifying the arithmetic expressions.

$1 y^{2} x^{2} y + x \times (y^{2} (\frac{d}{d x} [2] \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y] = x^{2} y^{3} + x \times (y^{2} (\frac{d}{d x} [2] \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y]$

The derivative of a constant value is always zero.

$x^{2} y^{3} + x \times (y^{2} (\frac{d}{d x} [2] \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y] = x^{2} y^{3} + x \times (y^{2} (0 \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y]$

Multiplying a number by zero always results in zero.

$x^{2} y^{3} + x \times (y^{2} (0 \times \ln (y) + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y] = x^{2} y^{3} + x \times (y^{2} (0 + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y]$

Simplifying the arithmetic expressions.

$x^{2} y^{3} + x \times (y^{2} (0 + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + x y^{2} \times (2 x) \times y + x y^{2} x^{2} \times \frac{d}{d x} [y] = x^{2} y^{3} + x \times (y^{2} (0 + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + 2 x^{2} y^{3} + x y^{2} x^{2} \times \frac{d}{d x} [y]$

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

$x^{2} y^{3} + x \times (y^{2} (0 + \frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + 2 x^{2} y^{3} + x y^{2} x^{2} \times \frac{d}{d x} [y] = x^{2} y^{3} + x \times (y^{2} \times (\frac{2}{y} \times \frac{d}{d x} [y])) \times x^{2} y + 2 x^{2} y^{3} + x y^{2} x^{2} \times \frac{d}{d x} [y]$

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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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