Calculus Examples

$\sin (4 x^{2} + 3 x)$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $4 x^{2} + 3 x$ .

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

Step 1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

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

Step 1.3

Replace all occurrences of $u$ with $4 x^{2} + 3 x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{2} + 3 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [3 x]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $4$ .

$\cos (4 x^{2} + 3 x) (8 x + \frac{d}{d x} [3 x])$

Step 2.5

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

$\cos (4 x^{2} + 3 x) (8 x + 3 \frac{d}{d x} [x])$

Step 2.6

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

$\cos (4 x^{2} + 3 x) (8 x + 3 \cdot 1)$

Step 2.7

Multiply $3$ by $1$ .

$\cos (4 x^{2} + 3 x) (8 x + 3)$