Calculus Examples

$y = \cos (2 x - 6)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x - 6$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $2 x - 6$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [2 x - 6]$

Step 1.1.2

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

$- \sin (u) \frac{d}{d x} [2 x - 6]$

Step 1.1.3

Replace all occurrences of $u$ with $2 x - 6$ .

$- \sin (2 x - 6) \frac{d}{d x} [2 x - 6]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $2 x - 6$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6]$ .

$- \sin (2 x - 6) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6])$

Step 1.2.2

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

$- \sin (2 x - 6) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 6])$

Step 1.2.3

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

$- \sin (2 x - 6) (2 \cdot 1 + \frac{d}{d x} [- 6])$

Step 1.2.4

Multiply $2$ by $1$ .

$- \sin (2 x - 6) (2 + \frac{d}{d x} [- 6])$

Step 1.2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$- \sin (2 x - 6) (2 + 0)$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2$ and $0$ .

$- \sin (2 x - 6) \cdot 2$

Step 1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = - 2 \sin (2 x - 6)$

Step 2

Find the second derivative.

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Step 2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \sin (2 x - 6)$ with respect to $x$ is $- 2 \frac{d}{d x} [\sin (2 x - 6)]$ .

$- 2 \frac{d}{d x} [\sin (2 x - 6)]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x - 6$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $2 x - 6$ .

$- 2 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x - 6])$

Step 2.2.2

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

$- 2 (\cos (u) \frac{d}{d x} [2 x - 6])$

Step 2.2.3

Replace all occurrences of $u$ with $2 x - 6$ .

$- 2 (\cos (2 x - 6) \frac{d}{d x} [2 x - 6])$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $2 x - 6$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6]$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $2$ by $1$ .

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

Step 2.3.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$- 2 \cos (2 x - 6) (2 + 0)$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $2$ and $0$ .

$- 2 \cos (2 x - 6) \cdot 2$

Step 2.3.6.2

Multiply $2$ by $- 2$ .

$f'' (x) = - 4 \cos (2 x - 6)$