Calculus Examples

$f (x) = \frac{\cos (8 x) - 1}{\sin (9 x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \cos (8 x) - 1$ and $g (x) = \sin (9 x)$ .

$\frac{\sin (9 x) \frac{d}{d x} [\cos (8 x) - 1] - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 2

By the Sum Rule, the derivative of $\cos (8 x) - 1$ with respect to $x$ is $\frac{d}{d x} [\cos (8 x)] + \frac{d}{d x} [- 1]$ .

$\frac{\sin (9 x) (\frac{d}{d x} [\cos (8 x)] + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 3

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 $g (x) = 8 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $8 x$ .

$\frac{\sin (9 x) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [8 x] + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 3.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$\frac{\sin (9 x) (- \sin (u_{1}) \frac{d}{d x} [8 x] + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 3.3

Replace all occurrences of $u_{1}$ with $8 x$ .

$\frac{\sin (9 x) (- \sin (8 x) \frac{d}{d x} [8 x] + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 4

Differentiate.

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Step 4.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{\sin (9 x) (- \sin (8 x) (8 \frac{d}{d x} [x]) + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 4.2

Multiply $8$ by $- 1$ .

$\frac{\sin (9 x) (- 8 \sin (8 x) \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 4.3

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

$\frac{\sin (9 x) (- 8 \sin (8 x) \cdot 1 + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 4.4

Multiply $- 8$ by $1$ .

$\frac{\sin (9 x) (- 8 \sin (8 x) + \frac{d}{d x} [- 1]) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 4.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{\sin (9 x) (- 8 \sin (8 x) + 0) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 4.6

Add $- 8 \sin (8 x)$ and $0$ .

$\frac{\sin (9 x) (- 8 \sin (8 x)) - (\cos (8 x) - 1) \frac{d}{d x} [\sin (9 x)]}{\sin^{2} (9 x)}$

Step 5

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 $g (x) = 9 x$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $9 x$ .

$\frac{\sin (9 x) (- 8 \sin (8 x)) - (\cos (8 x) - 1) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [9 x])}{\sin^{2} (9 x)}$

Step 5.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\frac{\sin (9 x) (- 8 \sin (8 x)) - (\cos (8 x) - 1) (\cos (u_{2}) \frac{d}{d x} [9 x])}{\sin^{2} (9 x)}$

Step 5.3

Replace all occurrences of $u_{2}$ with $9 x$ .

$\frac{\sin (9 x) (- 8 \sin (8 x)) - (\cos (8 x) - 1) (\cos (9 x) \frac{d}{d x} [9 x])}{\sin^{2} (9 x)}$

Step 6

Differentiate.

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Step 6.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$\frac{\sin (9 x) (- 8 \sin (8 x)) - (\cos (8 x) - 1) \cos (9 x) (9 \frac{d}{d x} [x])}{\sin^{2} (9 x)}$

Step 6.2

Multiply $9$ by $- 1$ .

$\frac{\sin (9 x) (- 8 \sin (8 x)) - 9 (\cos (8 x) - 1) \cos (9 x) \frac{d}{d x} [x]}{\sin^{2} (9 x)}$

Step 6.3

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

$\frac{\sin (9 x) (- 8 \sin (8 x)) - 9 (\cos (8 x) - 1) \cos (9 x) \cdot 1}{\sin^{2} (9 x)}$

Step 6.4

Multiply $- 9$ by $1$ .

$\frac{\sin (9 x) (- 8 \sin (8 x)) - 9 (\cos (8 x) - 1) \cos (9 x)}{\sin^{2} (9 x)}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{\sin (9 x) (- 8 \sin (8 x)) + (- 9 \cos (8 x) - 9 \cdot - 1) \cos (9 x)}{\sin^{2} (9 x)}$

Step 7.2

Apply the distributive property.

$\frac{\sin (9 x) (- 8 \sin (8 x)) - 9 \cos (8 x) \cos (9 x) - 9 \cdot - 1 \cos (9 x)}{\sin^{2} (9 x)}$

Step 7.3

Simplify each term.

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Step 7.3.1

Rewrite using the commutative property of multiplication.

$\frac{- 8 \sin (9 x) \sin (8 x) - 9 \cos (8 x) \cos (9 x) - 9 \cdot - 1 \cos (9 x)}{\sin^{2} (9 x)}$

Step 7.3.2

Multiply $- 9$ by $- 1$ .

$\frac{- 8 \sin (9 x) \sin (8 x) - 9 \cos (8 x) \cos (9 x) + 9 \cos (9 x)}{\sin^{2} (9 x)}$

Step 7.4

Reorder terms.

$\frac{- 8 \sin (8 x) \sin (9 x) - 9 \cos (8 x) \cos (9 x) + 9 \cos (9 x)}{\sin^{2} (9 x)}$