Calculus Examples

$5 \cos (7 \ln (x))$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (7 \ln (x))$ with respect to $x$ is $5 \frac{d}{d x} [\cos (7 \ln (x))]$ .

$5 \frac{d}{d x} [\cos (7 \ln (x))]$

Step 2

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 $g (x) = 7 \ln (x)$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $7 \ln (x)$ .

$5 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [7 \ln (x)])$

Step 2.2

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

$5 (- \sin (u) \frac{d}{d x} [7 \ln (x)])$

Step 2.3

Replace all occurrences of $u$ with $7 \ln (x)$ .

$5 (- \sin (7 \ln (x)) \frac{d}{d x} [7 \ln (x)])$

Step 3

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $5$ .

$- 5 (\sin (7 \ln (x)) \frac{d}{d x} [7 \ln (x)])$

Step 3.2

Since $7$ is constant with respect to $x$ , the derivative of $7 \ln (x)$ with respect to $x$ is $7 \frac{d}{d x} [\ln (x)]$ .

$- 5 \sin (7 \ln (x)) (7 \frac{d}{d x} [\ln (x)])$

Step 3.3

Multiply $7$ by $- 5$ .

$- 35 \sin (7 \ln (x)) \frac{d}{d x} [\ln (x)]$

Step 4

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

$- 35 \sin (7 \ln (x)) \frac{1}{x}$

Step 5

Combine fractions.

Tap for more steps...

Step 5.1

Combine $\frac{1}{x}$ and $- 35$ .

$\frac{- 35}{x} \sin (7 \ln (x))$

Step 5.2

Combine $\frac{- 35}{x}$ and $\sin (7 \ln (x))$ .

$\frac{- 35 \sin (7 \ln (x))}{x}$

Step 5.3

Move the negative in front of the fraction.

$- \frac{35 \sin (7 \ln (x))}{x}$

Step 6

Simplify $7 \ln (x)$ by moving $7$ inside the logarithm.

$- \frac{35 \sin (\ln (x^{7}))}{x}$