Calculus Examples

$y = {(3 x + 7)}^{5}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = 3 x + 7$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x + 7$ .

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [3 x + 7]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [3 x + 7]$

Step 1.3

Replace all occurrences of $u$ with $3 x + 7$ .

$5 {(3 x + 7)}^{4} \frac{d}{d x} [3 x + 7]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x + 7$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [7]$ .

$5 {(3 x + 7)}^{4} (\frac{d}{d x} [3 x] + \frac{d}{d x} [7])$

Step 2.2

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

$5 {(3 x + 7)}^{4} (3 \frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 2.3

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

$5 {(3 x + 7)}^{4} (3 \cdot 1 + \frac{d}{d x} [7])$

Step 2.4

Multiply $3$ by $1$ .

$5 {(3 x + 7)}^{4} (3 + \frac{d}{d x} [7])$

Step 2.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$5 {(3 x + 7)}^{4} (3 + 0)$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

$5 {(3 x + 7)}^{4} \cdot 3$

Step 2.6.2

Multiply $3$ by $5$ .

$15 {(3 x + 7)}^{4}$