Calculus Examples

$f (x) = 9 \cos (9 x) (10 \cos (10 x))$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Multiply $10$ by $9$ .

$\frac{d}{d x} [90 \cos (9 x) \cos (10 x)]$

Step 1.2

Since $90$ is constant with respect to $x$ , the derivative of $90 \cos (9 x) \cos (10 x)$ with respect to $x$ is $90 \frac{d}{d x} [\cos (9 x) \cos (10 x)]$ .

$90 \frac{d}{d x} [\cos (9 x) \cos (10 x)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos (9 x)$ and $g (x) = \cos (10 x)$ .

$90 (\cos (9 x) \frac{d}{d x} [\cos (10 x)] + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

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 $g (x) = 10 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $10 x$ .

$90 (\cos (9 x) (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [10 x]) + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

Step 3.2

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

$90 (\cos (9 x) (- \sin (u_{1}) \frac{d}{d x} [10 x]) + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

Step 3.3

Replace all occurrences of $u_{1}$ with $10 x$ .

$90 (\cos (9 x) (- \sin (10 x) \frac{d}{d x} [10 x]) + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

Step 4

Differentiate.

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Step 4.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$90 (\cos (9 x) (- \sin (10 x) (10 \frac{d}{d x} [x])) + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

Step 4.2

Multiply $10$ by $- 1$ .

$90 (\cos (9 x) (- 10 \sin (10 x) \frac{d}{d x} [x]) + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

Step 4.3

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

$90 (\cos (9 x) (- 10 \sin (10 x) \cdot 1) + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

Step 4.4

Multiply $- 10$ by $1$ .

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) \frac{d}{d x} [\cos (9 x)])$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 9 x$ .

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Step 5.1

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

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [9 x]))$

Step 5.2

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

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) (- \sin (u_{2}) \frac{d}{d x} [9 x]))$

Step 5.3

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

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) (- \sin (9 x) \frac{d}{d x} [9 x]))$

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 $9 \frac{d}{d x} [x]$ .

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) (- \sin (9 x) (9 \frac{d}{d x} [x])))$

Step 6.2

Multiply $9$ by $- 1$ .

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) (- 9 \sin (9 x) \frac{d}{d x} [x]))$

Step 6.3

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

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) (- 9 \sin (9 x) \cdot 1))$

Step 6.4

Multiply $- 9$ by $1$ .

$90 (\cos (9 x) (- 10 \sin (10 x)) + \cos (10 x) (- 9 \sin (9 x)))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$90 (\cos (9 x) (- 10 \sin (10 x))) + 90 (\cos (10 x) (- 9 \sin (9 x)))$

Step 7.2

Combine terms.

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Step 7.2.1

Multiply $- 10$ by $90$ .

$- 900 (\cos (9 x) (\sin (10 x))) + 90 (\cos (10 x) (- 9 \sin (9 x)))$

Step 7.2.2

Multiply $- 9$ by $90$ .

$- 900 \cos (9 x) \sin (10 x) - 810 \cos (10 x) \sin (9 x)$