Calculus Examples

$r = (2 {(\tan^{3} (2 x - 1))}^{\frac{1}{2}})$

Step 1

Rewrite the right side with rational exponents.

Tap for more steps...

Step 1.1

Remove parentheses.

$r = 2 {(\tan^{3} (2 x - 1))}^{\frac{1}{2}}$

Step 1.2

Multiply the exponents in ${(\tan^{3} (2 x - 1))}^{\frac{1}{2}}$ .

Tap for more steps...

Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = 2 {\tan (2 x - 1)}^{3 (\frac{1}{2})}$

Step 1.2.2

Combine $3$ and $\frac{1}{2}$ .

$r = 2 {\tan (2 x - 1)}^{\frac{3}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (r) = \frac{d}{d x} (2 {\tan (2 x - 1)}^{\frac{3}{2}})$

Step 3

The derivative of $r$ with respect to $x$ is $r'$ .

$r'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 {\tan (2 x - 1)}^{\frac{3}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [{\tan (2 x - 1)}^{\frac{3}{2}}]$ .

$2 \frac{d}{d x} [{\tan (2 x - 1)}^{\frac{3}{2}}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = \tan (2 x - 1)$ .

Tap for more steps...

Step 4.2.1

To apply the Chain Rule, set $u_{1}$ as $\tan (2 x - 1)$ .

$2 (\frac{d}{d u_{1}} [{u_{1}}^{\frac{3}{2}}] \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{3}{2}$ .

$2 (\frac{3}{2} {u_{1}}^{\frac{3}{2} - 1} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.2.3

Replace all occurrences of $u_{1}$ with $\tan (2 x - 1)$ .

$2 (\frac{3}{2} {\tan (2 x - 1)}^{\frac{3}{2} - 1} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 (\frac{3}{2} {\tan (2 x - 1)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.4

Combine $- 1$ and $\frac{2}{2}$ .

$2 (\frac{3}{2} {\tan (2 x - 1)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.5

Combine the numerators over the common denominator.

$2 (\frac{3}{2} {\tan (2 x - 1)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Multiply $- 1$ by $2$ .

$2 (\frac{3}{2} {\tan (2 x - 1)}^{\frac{3 - 2}{2}} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.6.2

Subtract $2$ from $3$ .

$2 (\frac{3}{2} {\tan (2 x - 1)}^{\frac{1}{2}} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.7

Combine $\frac{3}{2}$ and ${\tan (2 x - 1)}^{\frac{1}{2}}$ .

$2 (\frac{3 {\tan (2 x - 1)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\tan (2 x - 1)])$

Step 4.8

Combine $\frac{3 {\tan (2 x - 1)}^{\frac{1}{2}}}{2}$ and $2$ .

$\frac{3 {\tan (2 x - 1)}^{\frac{1}{2}} \cdot 2}{2} \frac{d}{d x} [\tan (2 x - 1)]$

Step 4.9

Multiply $2$ by $3$ .

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

Step 4.10

Factor $2$ out of $6 {\tan (2 x - 1)}^{\frac{1}{2}}$ .

$\frac{2 (3 {\tan (2 x - 1)}^{\frac{1}{2}})}{2} \frac{d}{d x} [\tan (2 x - 1)]$

Step 4.11

Cancel the common factors.

Tap for more steps...

Step 4.11.1

Factor $2$ out of $2$ .

$\frac{2 (3 {\tan (2 x - 1)}^{\frac{1}{2}})}{2 (1)} \frac{d}{d x} [\tan (2 x - 1)]$

Step 4.11.2

Cancel the common factor.

$\frac{2 (3 {\tan (2 x - 1)}^{\frac{1}{2}})}{2 \cdot 1} \frac{d}{d x} [\tan (2 x - 1)]$

Step 4.11.3

Rewrite the expression.

$\frac{3 {\tan (2 x - 1)}^{\frac{1}{2}}}{1} \frac{d}{d x} [\tan (2 x - 1)]$

Step 4.11.4

Divide $3 {\tan (2 x - 1)}^{\frac{1}{2}}$ by $1$ .

$3 {\tan (2 x - 1)}^{\frac{1}{2}} \frac{d}{d x} [\tan (2 x - 1)]$

Step 4.12

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 2 x - 1$ .

Tap for more steps...

Step 4.12.1

To apply the Chain Rule, set $u_{2}$ as $2 x - 1$ .

$3 {\tan (2 x - 1)}^{\frac{1}{2}} (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [2 x - 1])$

Step 4.12.2

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

$3 {\tan (2 x - 1)}^{\frac{1}{2}} (\sec^{2} (u_{2}) \frac{d}{d x} [2 x - 1])$

Step 4.12.3

Replace all occurrences of $u_{2}$ with $2 x - 1$ .

$3 {\tan (2 x - 1)}^{\frac{1}{2}} (\sec^{2} (2 x - 1) \frac{d}{d x} [2 x - 1])$

Step 4.13

Differentiate.

Tap for more steps...

Step 4.13.1

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$3 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1])$

Step 4.13.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$3 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 4.13.3

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

$3 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1) (2 \cdot 1 + \frac{d}{d x} [- 1])$

Step 4.13.4

Multiply $2$ by $1$ .

$3 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1) (2 + \frac{d}{d x} [- 1])$

Step 4.13.5

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

$3 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1) (2 + 0)$

Step 4.13.6

Simplify the expression.

Tap for more steps...

Step 4.13.6.1

Add $2$ and $0$ .

$3 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1) \cdot 2$

Step 4.13.6.2

Multiply $2$ by $3$ .

$6 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1)$

Step 5

Reform the equation by setting the left side equal to the right side.

$r' = 6 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1)$

Step 6

Replace $r'$ with $\frac{d r}{d x}$ .

$\frac{d r}{d x} = 6 {\tan (2 x - 1)}^{\frac{1}{2}} \sec^{2} (2 x - 1)$