Calculus Examples

$y = - 5 {(2 x + 3)}^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (- 5 {(2 x + 3)}^{3})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

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 $f (x) = x^{3}$ and $g (x) = 2 x + 3$ .

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Step 3.2.1

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

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

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

Step 3.2.3

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

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $3$ by $- 5$ .

$- 15 ({(2 x + 3)}^{2} \frac{d}{d x} [2 x + 3])$

Step 3.3.2

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

$- 15 {(2 x + 3)}^{2} (\frac{d}{d x} [2 x] + \frac{d}{d x} [3])$

Step 3.3.3

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

$- 15 {(2 x + 3)}^{2} (2 \frac{d}{d x} [x] + \frac{d}{d x} [3])$

Step 3.3.4

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

$- 15 {(2 x + 3)}^{2} (2 \cdot 1 + \frac{d}{d x} [3])$

Step 3.3.5

Multiply $2$ by $1$ .

$- 15 {(2 x + 3)}^{2} (2 + \frac{d}{d x} [3])$

Step 3.3.6

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

$- 15 {(2 x + 3)}^{2} (2 + 0)$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add $2$ and $0$ .

$- 15 {(2 x + 3)}^{2} \cdot 2$

Step 3.3.7.2

Multiply $2$ by $- 15$ .

$- 30 {(2 x + 3)}^{2}$

Step 4

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

$y' = - 30 {(2 x + 3)}^{2}$

Step 5

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

$\frac{d y}{d x} = - 30 {(2 x + 3)}^{2}$