Calculus Examples

$z = y {(3 + 2 x)}^{4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (z) = \frac{d}{d x} (y {(3 + 2 x)}^{4})$

Step 2

The derivative of $z$ with respect to $x$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $y$ is constant with respect to $x$ , the derivative of $y {(3 + 2 x)}^{4}$ with respect to $x$ is $y \frac{d}{d x} [{(3 + 2 x)}^{4}]$ .

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 3 + 2 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $3 + 2 x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $3 + 2 x$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

Move $4$ to the left of $y$ .

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

Step 3.3.2

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

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

Step 3.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 3.3.4

Add $0$ and $\frac{d}{d x} [2 x]$ .

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

Step 3.3.5

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

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

Step 3.3.6

Multiply $2$ by $4$ .

$8 y {(3 + 2 x)}^{3} \frac{d}{d x} [x]$

Step 3.3.7

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

$8 y {(3 + 2 x)}^{3} \cdot 1$

Step 3.3.8

Multiply $8$ by $1$ .

$8 y {(3 + 2 x)}^{3}$

Step 4

Reform the equation by setting the left side equal to the right side.

$z' = 8 y {(3 + 2 x)}^{3}$

Step 5

Replace $z'$ with $\frac{d z}{d x}$ .

$\frac{d z}{d x} = 8 y {(3 + 2 x)}^{3}$