Calculus Examples

$y = \sqrt{1 + \tan^{2} (x)}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + \tan^{2} (x)}$ as ${(1 + \tan^{2} (x))}^{\frac{1}{2}}$ .

$y = {(1 + \tan^{2} (x))}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(1 + \tan^{2} (x))}^{\frac{1}{2}})$

Step 3

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

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

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

Step 4.5.2

Subtract $2$ from $1$ .

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

Step 4.6

Differentiate.

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Step 4.6.1

Move the negative in front of the fraction.

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

Step 4.6.2

Combine fractions.

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Step 4.6.2.1

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

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

Step 4.6.2.2

Move ${(1 + \tan^{2} (x))}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.6.3

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

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

Step 4.6.4

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

$\frac{1}{2 {(1 + \tan^{2} (x))}^{\frac{1}{2}}} (0 + \frac{d}{d x} [\tan^{2} (x)])$

Step 4.6.5

Add $0$ and $\frac{d}{d x} [\tan^{2} (x)]$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.8

Simplify terms.

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Step 4.8.1

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

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

Step 4.8.2

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

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

Step 4.8.3

Move $2$ to the left of $\tan (x)$ .

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

Step 4.8.4

Cancel the common factor.

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

Step 4.8.5

Rewrite the expression.

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

Step 4.9

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

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

Step 4.10

Combine $\frac{\tan (x)}{{(1 + \tan^{2} (x))}^{\frac{1}{2}}}$ and $\sec^{2} (x)$ .

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

Step 4.11

Simplify.

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Step 4.11.1

Reorder terms.

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

Step 4.11.2

Rearrange terms.

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

Step 4.11.3

Apply pythagorean identity.

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

Step 4.11.4

Simplify the denominator.

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Step 4.11.4.1

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

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Step 4.11.4.1.1

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

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

Step 4.11.4.1.2

Cancel the common factor of $2$ .

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Step 4.11.4.1.2.1

Cancel the common factor.

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

Step 4.11.4.1.2.2

Rewrite the expression.

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

Step 4.11.4.2

Simplify.

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

Step 4.11.5

Cancel the common factor of $\sec^{2} (x)$ and $\sec (x)$ .

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Step 4.11.5.1

Factor $\sec (x)$ out of $\sec^{2} (x) \tan (x)$ .

$\frac{\sec (x) (\sec (x) \tan (x))}{\sec (x)}$

Step 4.11.5.2

Cancel the common factors.

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Step 4.11.5.2.1

Multiply by $1$ .

$\frac{\sec (x) (\sec (x) \tan (x))}{\sec (x) \cdot 1}$

Step 4.11.5.2.2

Cancel the common factor.

$\frac{\sec (x) (\sec (x) \tan (x))}{\sec (x) \cdot 1}$

Step 4.11.5.2.3

Rewrite the expression.

$\frac{\sec (x) \tan (x)}{1}$

Step 4.11.5.2.4

Divide $\sec (x) \tan (x)$ by $1$ .

$\sec (x) \tan (x)$

Step 5

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

$y' = \sec (x) \tan (x)$

Step 6

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

$\frac{d y}{d x} = \sec (x) \tan (x)$