Calculus Examples

$f (x) = 10^{x \cos (x)}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 10^{x}$ and $g (x) = x \cos (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x \cos (x)$ .

$\frac{d}{d u} [10^{u}] \frac{d}{d x} [x \cos (x)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $10$ .

$10^{u} \ln (10) \frac{d}{d x} [x \cos (x)]$

Step 1.3

Replace all occurrences of $u$ with $x \cos (x)$ .

$10^{x \cos (x)} \ln (10) \frac{d}{d x} [x \cos (x)]$

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 $f (x) = x$ and $g (x) = \cos (x)$ .

$10^{x \cos (x)} \ln (10) (x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x])$

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$10^{x \cos (x)} \ln (10) (x (- \sin (x)) + \cos (x) \cdot 1)$

Step 4.2

Multiply $\cos (x)$ by $1$ .

$10^{x \cos (x)} \ln (10) (x (- \sin (x)) + \cos (x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$10^{x \cos (x)} \ln (10) (x (- \sin (x))) + 10^{x \cos (x)} \ln (10) \cos (x)$

Step 5.2

Reorder terms.

$- 10^{x \cos (x)} x \sin (x) \ln (10) + 10^{x \cos (x)} \cos (x) \ln (10)$