Calculus Examples

$2^{- \tan (3 x)}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 2^{x}$ and $g (x) = - \tan (3 x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $- \tan (3 x)$ .

$\frac{d}{d u_{1}} [2^{u_{1}}] \frac{d}{d x} [- \tan (3 x)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $2$ .

$2^{u_{1}} \ln (2) \frac{d}{d x} [- \tan (3 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $- \tan (3 x)$ .

$2^{- \tan (3 x)} \ln (2) \frac{d}{d x} [- \tan (3 x)]$

Step 2

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (3 x)$ with respect to $x$ is $- \frac{d}{d x} [\tan (3 x)]$ .

$2^{- \tan (3 x)} \ln (2) (- \frac{d}{d x} [\tan (3 x)])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 3 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$2^{- \tan (3 x)} \ln (2) (- (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [3 x]))$

Step 3.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$2^{- \tan (3 x)} \ln (2) (- (\sec^{2} (u_{2}) \frac{d}{d x} [3 x]))$

Step 3.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$2^{- \tan (3 x)} \ln (2) (- (\sec^{2} (3 x) \frac{d}{d x} [3 x]))$

Step 4

Differentiate.

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Step 4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$2^{- \tan (3 x)} \ln (2) (- \sec^{2} (3 x) (3 \frac{d}{d x} [x]))$

Step 4.2

Multiply $3$ by $- 1$ .

$2^{- \tan (3 x)} \ln (2) (- 3 \sec^{2} (3 x) \frac{d}{d x} [x])$

Step 4.3

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

$2^{- \tan (3 x)} \ln (2) (- 3 \sec^{2} (3 x) \cdot 1)$

Step 4.4

Simplify the expression.

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Step 4.4.1

Multiply $- 3$ by $1$ .

$2^{- \tan (3 x)} \ln (2) (- 3 \sec^{2} (3 x))$

Step 4.4.2

Reorder the factors of $2^{- \tan (3 x)} \ln (2) (- 3 \sec^{2} (3 x))$ .

$- 3 \cdot 2^{- \tan (3 x)} \sec^{2} (3 x) \ln (2)$