Calculus Examples

$4 y^{4} - 72 y^{2} + 9 x^{2} - 13$

Step 1

By the Sum Rule, the derivative of $4 y^{4} - 72 y^{2} + 9 x^{2} - 13$ with respect to $y$ is $\frac{d}{d y} [4 y^{4}] + \frac{d}{d y} [- 72 y^{2}] + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$ .

$\frac{d}{d y} [4 y^{4}] + \frac{d}{d y} [- 72 y^{2}] + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$

Step 2

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

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Step 2.1

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

$4 \frac{d}{d y} [y^{4}] + \frac{d}{d y} [- 72 y^{2}] + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$

Step 2.2

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

$4 (4 y^{3}) + \frac{d}{d y} [- 72 y^{2}] + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$

Step 2.3

Multiply $4$ by $4$ .

$16 y^{3} + \frac{d}{d y} [- 72 y^{2}] + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$

Step 3

Evaluate $\frac{d}{d y} [- 72 y^{2}]$ .

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Step 3.1

Since $- 72$ is constant with respect to $y$ , the derivative of $- 72 y^{2}$ with respect to $y$ is $- 72 \frac{d}{d y} [y^{2}]$ .

$16 y^{3} - 72 \frac{d}{d y} [y^{2}] + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$

Step 3.2

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

$16 y^{3} - 72 (2 y) + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$

Step 3.3

Multiply $2$ by $- 72$ .

$16 y^{3} - 144 y + \frac{d}{d y} [9 x^{2}] + \frac{d}{d y} [- 13]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $9 x^{2}$ is constant with respect to $y$ , the derivative of $9 x^{2}$ with respect to $y$ is $0$ .

$16 y^{3} - 144 y + 0 + \frac{d}{d y} [- 13]$

Step 4.2

Since $- 13$ is constant with respect to $y$ , the derivative of $- 13$ with respect to $y$ is $0$ .

$16 y^{3} - 144 y + 0 + 0$

Step 5

Combine terms.

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Step 5.1

Add $16 y^{3} - 144 y$ and $0$ .

$16 y^{3} - 144 y + 0$

Step 5.2

Add $16 y^{3} - 144 y$ and $0$ .

$16 y^{3} - 144 y$