Calculus Examples

$x^{4} - 5 x^{3} y^{2} + 2 y^{4}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{4} - 5 x^{3} y^{2} + 2 y^{4}$ with respect to $y$ is $\frac{d}{d y} [x^{4}] + \frac{d}{d y} [- 5 x^{3} y^{2}] + \frac{d}{d y} [2 y^{4}]$ .

$\frac{d}{d y} [x^{4}] + \frac{d}{d y} [- 5 x^{3} y^{2}] + \frac{d}{d y} [2 y^{4}]$

Step 1.2

Since $x^{4}$ is constant with respect to $y$ , the derivative of $x^{4}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- 5 x^{3} y^{2}] + \frac{d}{d y} [2 y^{4}]$

Step 2

Evaluate $\frac{d}{d y} [- 5 x^{3} y^{2}]$ .

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Step 2.1

Since $- 5 x^{3}$ is constant with respect to $y$ , the derivative of $- 5 x^{3} y^{2}$ with respect to $y$ is $- 5 x^{3} \frac{d}{d y} [y^{2}]$ .

$0 - 5 x^{3} \frac{d}{d y} [y^{2}] + \frac{d}{d y} [2 y^{4}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$0 - 5 x^{3} (2 y) + \frac{d}{d y} [2 y^{4}]$

Step 2.3

Multiply $2$ by $- 5$ .

$0 - 10 x^{3} y + \frac{d}{d y} [2 y^{4}]$

Step 3

Evaluate $\frac{d}{d y} [2 y^{4}]$ .

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Step 3.1

Since $2$ is constant with respect to $y$ , the derivative of $2 y^{4}$ with respect to $y$ is $2 \frac{d}{d y} [y^{4}]$ .

$0 - 10 x^{3} y + 2 \frac{d}{d y} [y^{4}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$0 - 10 x^{3} y + 2 (4 y^{3})$

Step 3.3

Multiply $4$ by $2$ .

$0 - 10 x^{3} y + 8 y^{3}$

Step 4

Simplify.

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Step 4.1

Subtract $10 x^{3} y$ from $0$ .

$- 10 x^{3} y + 8 y^{3}$

Step 4.2

Reorder terms.

$8 y^{3} - 10 x^{3} y$