Calculus Examples

$\ln (6 x - \cos (x))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 6 x - \cos (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $6 x - \cos (x)$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [6 x - \cos (x)]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [6 x - \cos (x)]$

Step 1.3

Replace all occurrences of $u$ with $6 x - \cos (x)$ .

$\frac{1}{6 x - \cos (x)} \frac{d}{d x} [6 x - \cos (x)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $6 x - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{1}{6 x - \cos (x)} (\frac{d}{d x} [6 x] + \frac{d}{d x} [- \cos (x)])$

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{1}{6 x - \cos (x)} (6 \frac{d}{d x} [x] + \frac{d}{d x} [- \cos (x)])$

Step 2.3

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

$\frac{1}{6 x - \cos (x)} (6 \cdot 1 + \frac{d}{d x} [- \cos (x)])$

Step 2.4

Multiply $6$ by $1$ .

$\frac{1}{6 x - \cos (x)} (6 + \frac{d}{d x} [- \cos (x)])$

Step 2.5

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

$\frac{1}{6 x - \cos (x)} (6 - \frac{d}{d x} [\cos (x)])$

Step 3

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

$\frac{1}{6 x - \cos (x)} (6 - - \sin (x))$

Step 4

Multiply.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{6 x - \cos (x)} (6 + 1 \sin (x))$

Step 4.2

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

$\frac{1}{6 x - \cos (x)} (6 + \sin (x))$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $\frac{1}{6 x - \cos (x)} (6 + \sin (x))$ .

$(6 + \sin (x)) \frac{1}{6 x - \cos (x)}$

Step 5.2

Apply the distributive property.

$6 \frac{1}{6 x - \cos (x)} + \sin (x) \frac{1}{6 x - \cos (x)}$

Step 5.3

Combine $6$ and $\frac{1}{6 x - \cos (x)}$ .

$\frac{6}{6 x - \cos (x)} + \sin (x) \frac{1}{6 x - \cos (x)}$

Step 5.4

Combine $\sin (x)$ and $\frac{1}{6 x - \cos (x)}$ .

$\frac{6}{6 x - \cos (x)} + \frac{\sin (x)}{6 x - \cos (x)}$

Step 5.5

Combine the numerators over the common denominator.

$\frac{6 + \sin (x)}{6 x - \cos (x)}$