Calculus Examples

$y = \log (1 - 4 \tan (x))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = 1 - 4 \tan (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $1 - 4 \tan (x)$ .

$\frac{d}{d u} [\log (u)] \frac{d}{d x} [1 - 4 \tan (x)]$

Step 1.2

The derivative of $\log (u)$ with respect to $u$ is $\frac{1}{u \ln (10)}$ .

$\frac{1}{u \ln (10)} \frac{d}{d x} [1 - 4 \tan (x)]$

Step 1.3

Replace all occurrences of $u$ with $1 - 4 \tan (x)$ .

$\frac{1}{(1 - 4 \tan (x)) \ln (10)} \frac{d}{d x} [1 - 4 \tan (x)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 - 4 \tan (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 \tan (x)]$ .

$\frac{1}{(1 - 4 \tan (x)) \ln (10)} (\frac{d}{d x} [1] + \frac{d}{d x} [- 4 \tan (x)])$

Step 2.2

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

$\frac{1}{(1 - 4 \tan (x)) \ln (10)} (0 + \frac{d}{d x} [- 4 \tan (x)])$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 4 \tan (x)]$ .

$\frac{1}{(1 - 4 \tan (x)) \ln (10)} \frac{d}{d x} [- 4 \tan (x)]$

Step 2.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \tan (x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\tan (x)]$ .

$\frac{1}{(1 - 4 \tan (x)) \ln (10)} (- 4 \frac{d}{d x} [\tan (x)])$

Step 2.5

Combine fractions.

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Step 2.5.1

Combine $- 4$ and $\frac{1}{(1 - 4 \tan (x)) \ln (10)}$ .

$\frac{- 4}{(1 - 4 \tan (x)) \ln (10)} \frac{d}{d x} [\tan (x)]$

Step 2.5.2

Move the negative in front of the fraction.

$- \frac{4}{(1 - 4 \tan (x)) \ln (10)} \frac{d}{d x} [\tan (x)]$

Step 3

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

$- \frac{4}{(1 - 4 \tan (x)) \ln (10)} \sec^{2} (x)$

Step 4

Combine fractions.

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Step 4.1

Combine $\sec^{2} (x)$ and $\frac{4}{(1 - 4 \tan (x)) \ln (10)}$ .

$- \frac{\sec^{2} (x) \cdot 4}{(1 - 4 \tan (x)) \ln (10)}$

Step 4.2

Move $4$ to the left of $\sec^{2} (x)$ .

$- \frac{4 \cdot \sec^{2} (x)}{(1 - 4 \tan (x)) \ln (10)}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- \frac{4 \sec^{2} (x)}{1 \ln (10) - 4 \tan (x) \ln (10)}$

Step 5.2

Multiply $\ln (10)$ by $1$ .

$- \frac{4 \sec^{2} (x)}{\ln (10) - 4 \tan (x) \ln (10)}$