Precalculus Examples

$y = {(5 x^{2} + 3)}^{4}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 5 x^{2} + 3$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $5 x^{2} + 3$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $5 x^{2} + 3$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $5$ .

$4 {(5 x^{2} + 3)}^{3} (10 x + \frac{d}{d x} [3])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$4 {(5 x^{2} + 3)}^{3} (10 x + 0)$

Step 4.2

Add $10 x$ and $0$ .

$4 {(5 x^{2} + 3)}^{3} (10 x)$

Step 5

Simplify.

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Step 5.1

Multiply $10$ by $4$ .

$40 {(5 x^{2} + 3)}^{3} x$

Step 5.2

Reorder the factors of $40 {(5 x^{2} + 3)}^{3} x$ .

$40 x {(5 x^{2} + 3)}^{3}$