Calculus Examples

$8 \sin (- 0.2 x + 0.9) + 10$

Step 1

By the Sum Rule, the derivative of $8 \sin (- 0.2 x + 0.9) + 10$ with respect to $x$ is $\frac{d}{d x} [8 \sin (- 0.2 x + 0.9)] + \frac{d}{d x} [10]$ .

$\frac{d}{d x} [8 \sin (- 0.2 x + 0.9)] + \frac{d}{d x} [10]$

Step 2

Evaluate $\frac{d}{d x} [8 \sin (- 0.2 x + 0.9)]$ .

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \sin (- 0.2 x + 0.9)$ with respect to $x$ is $8 \frac{d}{d x} [\sin (- 0.2 x + 0.9)]$ .

$8 \frac{d}{d x} [\sin (- 0.2 x + 0.9)] + \frac{d}{d x} [10]$

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 $g (x) = - 0.2 x + 0.9$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $- 0.2 x + 0.9$ .

$8 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [- 0.2 x + 0.9]) + \frac{d}{d x} [10]$

Step 2.2.2

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

$8 (\cos (u) \frac{d}{d x} [- 0.2 x + 0.9]) + \frac{d}{d x} [10]$

Step 2.2.3

Replace all occurrences of $u$ with $- 0.2 x + 0.9$ .

$8 (\cos (- 0.2 x + 0.9) \frac{d}{d x} [- 0.2 x + 0.9]) + \frac{d}{d x} [10]$

Step 2.3

By the Sum Rule, the derivative of $- 0.2 x + 0.9$ with respect to $x$ is $\frac{d}{d x} [- 0.2 x] + \frac{d}{d x} [0.9]$ .

$8 (\cos (- 0.2 x + 0.9) (\frac{d}{d x} [- 0.2 x] + \frac{d}{d x} [0.9])) + \frac{d}{d x} [10]$

Step 2.4

Since $- 0.2$ is constant with respect to $x$ , the derivative of $- 0.2 x$ with respect to $x$ is $- 0.2 \frac{d}{d x} [x]$ .

$8 (\cos (- 0.2 x + 0.9) (- 0.2 \frac{d}{d x} [x] + \frac{d}{d x} [0.9])) + \frac{d}{d x} [10]$

Step 2.5

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

$8 (\cos (- 0.2 x + 0.9) (- 0.2 \cdot 1 + \frac{d}{d x} [0.9])) + \frac{d}{d x} [10]$

Step 2.6

Since $0.9$ is constant with respect to $x$ , the derivative of $0.9$ with respect to $x$ is $0$ .

$8 (\cos (- 0.2 x + 0.9) (- 0.2 \cdot 1 + 0)) + \frac{d}{d x} [10]$

Step 2.7

Multiply $- 0.2$ by $1$ .

$8 (\cos (- 0.2 x + 0.9) (- 0.2 + 0)) + \frac{d}{d x} [10]$

Step 2.8

Add $- 0.2$ and $0$ .

$8 (\cos (- 0.2 x + 0.9) \cdot - 0.2) + \frac{d}{d x} [10]$

Step 2.9

Move $- 0.2$ to the left of $\cos (- 0.2 x + 0.9)$ .

$8 (- 0.2 \cdot \cos (- 0.2 x + 0.9)) + \frac{d}{d x} [10]$

Step 2.10

Multiply $- 0.2$ by $8$ .

$- 1.6 \cos (- 0.2 x + 0.9) + \frac{d}{d x} [10]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$- 1.6 \cos (- 0.2 x + 0.9) + 0$

Step 3.2

Add $- 1.6 \cos (- 0.2 x + 0.9)$ and $0$ .

$- 1.6 \cos (- 0.2 x + 0.9)$