Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin^{-1} (x)$ and $g (x) = 6 x + 1$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $6 x + 1$ .

$\frac{d}{d u} [\sin^{-1} (u)] \frac{d}{d x} [6 x + 1]$

Step 1.2

The derivative of $\sin^{-1} (u)$ with respect to $u$ is $\frac{1}{\sqrt{1 - u^{2}}}$ .

$\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [6 x + 1]$

Step 1.3

Replace all occurrences of $u$ with $6 x + 1$ .

$\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}} \frac{d}{d x} [6 x + 1]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $6 x + 1$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [1]$ .

$\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}} (\frac{d}{d x} [6 x] + \frac{d}{d x} [1])$

Step 2.2

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

$\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}} (6 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.3

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

$\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}} (6 \cdot 1 + \frac{d}{d x} [1])$

Step 2.4

Multiply $6$ by $1$ .

$\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}} (6 + \frac{d}{d x} [1])$

Step 2.5

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

$\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}} (6 + 0)$

Step 2.6

Combine fractions.

Tap for more steps...

Step 2.6.1

Add $6$ and $0$ .

$\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}} \cdot 6$

Step 2.6.2

Combine $\frac{1}{\sqrt{1 - {(6 x + 1)}^{2}}}$ and $6$ .

$\frac{6}{\sqrt{1 - {(6 x + 1)}^{2}}}$