Calculus Examples

\cos (5 x)

$\cos (5 x)$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = \cos (x)

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

g (x) = 5 x

$g (x) = 5 x$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set

u

$u$ as

5 x

$5 x$ .

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

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

Step 1.2

The derivative of

\cos (u)

$\cos (u)$ with respect to

u

$u$ is

- \sin (u)

$- \sin (u)$ .

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

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

Step 1.3

Replace all occurrences of

u

$u$ with

5 x

$5 x$ .

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

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

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

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

5 x

$5 x$ with respect to

x

$x$ is

5 \frac{d}{d x} [x]

$5 \frac{d}{d x} [x]$ .

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

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

Step 2.2

Multiply

5

$5$ by

- 1

$- 1$ .

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

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

Step 2.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

- 5 \sin (5 x) \cdot 1

$- 5 \sin (5 x) \cdot 1$

Step 2.4

Multiply

- 5

$- 5$ by

1

$1$ .

- 5 \sin (5 x)

$- 5 \sin (5 x)$

- 5 \sin (5 x)

$- 5 \sin (5 x)$

\cos 5 x

$\cos 5 x$

(

)

[

]

√



≥



∫













≤

















∞













