Calculus Examples

$\sin (- 3 x^{2} - 3 x + 7)$

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 $g (x) = - 3 x^{2} - 3 x + 7$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $- 3 x^{2} - 3 x + 7$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [- 3 x^{2} - 3 x + 7]$

Step 1.2

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

$\cos (u) \frac{d}{d x} [- 3 x^{2} - 3 x + 7]$

Step 1.3

Replace all occurrences of $u$ with $- 3 x^{2} - 3 x + 7$ .

$\cos (- 3 x^{2} - 3 x + 7) \frac{d}{d x} [- 3 x^{2} - 3 x + 7]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $- 3 x^{2} - 3 x + 7$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [7]$ .

$\cos (- 3 x^{2} - 3 x + 7) (\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [7])$

Step 2.2

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

$\cos (- 3 x^{2} - 3 x + 7) (- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [7])$

Step 2.3

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

$\cos (- 3 x^{2} - 3 x + 7) (- 3 (2 x) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [7])$

Step 2.4

Multiply $2$ by $- 3$ .

$\cos (- 3 x^{2} - 3 x + 7) (- 6 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [7])$

Step 2.5

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

$\cos (- 3 x^{2} - 3 x + 7) (- 6 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 2.6

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

$\cos (- 3 x^{2} - 3 x + 7) (- 6 x - 3 \cdot 1 + \frac{d}{d x} [7])$

Step 2.7

Multiply $- 3$ by $1$ .

$\cos (- 3 x^{2} - 3 x + 7) (- 6 x - 3 + \frac{d}{d x} [7])$

Step 2.8

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

$\cos (- 3 x^{2} - 3 x + 7) (- 6 x - 3 + 0)$

Step 2.9

Add $- 6 x - 3$ and $0$ .

$\cos (- 3 x^{2} - 3 x + 7) (- 6 x - 3)$