Calculus Examples

$y = \frac{\sqrt{4 + 6 x^{3}}}{\cos^{3} (x)}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + 6 x^{3}}$ as ${(4 + 6 x^{3})}^{\frac{1}{2}}$ .

$y = \frac{{(4 + 6 x^{3})}^{\frac{1}{2}}}{\cos^{3} (x)}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{{(4 + 6 x^{3})}^{\frac{1}{2}}}{\cos^{3} (x)})$

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

Convert from $\frac{1}{\cos^{3} (x)}$ to $\sec^{3} (x)$ .

$\frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x)]$

Step 4.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(4 + 6 x^{3})}^{\frac{1}{2}}$ and $g (x) = \sec^{3} (x)$ .

${(4 + 6 x^{3})}^{\frac{1}{2}} \frac{d}{d x} [\sec^{3} (x)] + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.3

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

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Step 4.3.1

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

${(4 + 6 x^{3})}^{\frac{1}{2}} (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.3.2

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

${(4 + 6 x^{3})}^{\frac{1}{2}} (3 {u_{1}}^{2} \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.3.3

Replace all occurrences of $u_{1}$ with $\sec (x)$ .

${(4 + 6 x^{3})}^{\frac{1}{2}} (3 \sec^{2} (x) \frac{d}{d x} [\sec (x)]) + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.4

Move $3$ to the left of ${(4 + 6 x^{3})}^{\frac{1}{2}}$ .

$3 \cdot {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{2} (x) \frac{d}{d x} [\sec (x)] + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.5

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.6

Raise $\sec (x)$ to the power of $1$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} (\sec^{1} (x) \sec^{2} (x)) \tan (x) + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} {\sec (x)}^{1 + 2} \tan (x) + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.8

Add $1$ and $2$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) \frac{d}{d x} [{(4 + 6 x^{3})}^{\frac{1}{2}}]$

Step 4.9

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 $g (x) = 4 + 6 x^{3}$ .

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Step 4.9.1

To apply the Chain Rule, set $u_{2}$ as $4 + 6 x^{3}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.9.2

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

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.9.3

Replace all occurrences of $u_{2}$ with $4 + 6 x^{3}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {(4 + 6 x^{3})}^{\frac{1}{2} - 1} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.10

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

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {(4 + 6 x^{3})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.11

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

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {(4 + 6 x^{3})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.12

Combine the numerators over the common denominator.

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {(4 + 6 x^{3})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.13

Simplify the numerator.

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Step 4.13.1

Multiply $- 1$ by $2$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {(4 + 6 x^{3})}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.13.2

Subtract $2$ from $1$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {(4 + 6 x^{3})}^{\frac{- 1}{2}} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.14

Combine fractions.

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Step 4.14.1

Move the negative in front of the fraction.

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2} {(4 + 6 x^{3})}^{- \frac{1}{2}} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.14.2

Combine $\frac{1}{2}$ and ${(4 + 6 x^{3})}^{- \frac{1}{2}}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{{(4 + 6 x^{3})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.14.3

Move ${(4 + 6 x^{3})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \sec^{3} (x) (\frac{1}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [4 + 6 x^{3}])$

Step 4.14.4

Combine $\frac{1}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}}$ and $\sec^{3} (x)$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{\sec^{3} (x)}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [4 + 6 x^{3}]$

Step 4.15

By the Sum Rule, the derivative of $4 + 6 x^{3}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [6 x^{3}]$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{\sec^{3} (x)}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} (\frac{d}{d x} [4] + \frac{d}{d x} [6 x^{3}])$

Step 4.16

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{\sec^{3} (x)}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [6 x^{3}])$

Step 4.17

Add $0$ and $\frac{d}{d x} [6 x^{3}]$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{\sec^{3} (x)}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{3}]$

Step 4.18

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{\sec^{3} (x)}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} (6 \frac{d}{d x} [x^{3}])$

Step 4.19

Simplify terms.

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Step 4.19.1

Combine $6$ and $\frac{\sec^{3} (x)}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{6 \sec^{3} (x)}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [x^{3}]$

Step 4.19.2

Factor $2$ out of $6 \sec^{3} (x)$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{2 (3 \sec^{3} (x))}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [x^{3}]$

Step 4.20

Cancel the common factors.

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Step 4.20.1

Factor $2$ out of $2 {(4 + 6 x^{3})}^{\frac{1}{2}}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{2 (3 \sec^{3} (x))}{2 ({(4 + 6 x^{3})}^{\frac{1}{2}})} \frac{d}{d x} [x^{3}]$

Step 4.20.2

Cancel the common factor.

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{2 (3 \sec^{3} (x))}{2 {(4 + 6 x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [x^{3}]$

Step 4.20.3

Rewrite the expression.

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{3 \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}} \frac{d}{d x} [x^{3}]$

Step 4.21

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

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{3 \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}} (3 x^{2})$

Step 4.22

Combine fractions.

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Step 4.22.1

Combine $3$ and $\frac{3 \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{3 (3 \sec^{3} (x))}{{(4 + 6 x^{3})}^{\frac{1}{2}}} x^{2}$

Step 4.22.2

Multiply $3$ by $3$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{9 \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}} x^{2}$

Step 4.22.3

Combine $\frac{9 \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}}$ and $x^{2}$ .

$3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x) + \frac{9 \sec^{3} (x) x^{2}}{{(4 + 6 x^{3})}^{\frac{1}{2}}}$

Step 4.23

Simplify.

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Step 4.23.1

Reorder terms.

$\frac{9 \sec^{3} (x) x^{2}}{{(4 + 6 x^{3})}^{\frac{1}{2}}} + 3 \sec^{3} (x) {(4 + 6 x^{3})}^{\frac{1}{2}} \tan (x)$

Step 4.23.2

Reorder factors in $\frac{9 \sec^{3} (x) x^{2}}{{(4 + 6 x^{3})}^{\frac{1}{2}}} + 3 \sec^{3} (x) {(4 + 6 x^{3})}^{\frac{1}{2}} \tan (x)$ .

$\frac{9 x^{2} \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}} + 3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x)$

Step 5

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

$y' = \frac{9 x^{2} \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}} + 3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x)$

Step 6

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

$\frac{d y}{d x} = \frac{9 x^{2} \sec^{3} (x)}{{(4 + 6 x^{3})}^{\frac{1}{2}}} + 3 {(4 + 6 x^{3})}^{\frac{1}{2}} \sec^{3} (x) \tan (x)$