Calculus Examples

$y = e^{k \tan (\sqrt{5 x})}$

Step 1

Simplify with factoring out.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [e^{k \tan ({(5 x)}^{\frac{1}{2}})}]$

Step 1.2

Factor $5$ out of $5 x$ .

$\frac{d}{d x} [e^{k \tan ({(5 (x))}^{\frac{1}{2}})}]$

Step 1.3

Apply the product rule to $5 (x)$ .

$\frac{d}{d x} [e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})]$

Step 2.3

Replace all occurrences of $u_{1}$ with $k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \frac{d}{d x} [k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})]$

Step 3

Since $k$ is constant with respect to $x$ , the derivative of $k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})$ with respect to $x$ is $k \frac{d}{d x} [\tan (5^{\frac{1}{2}} x^{\frac{1}{2}})]$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} (k \frac{d}{d x} [\tan (5^{\frac{1}{2}} x^{\frac{1}{2}})])$

Step 4

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 $g (x) = 5^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $5^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [5^{\frac{1}{2}} x^{\frac{1}{2}}])$

Step 4.2

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

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k (\sec^{2} (u_{2}) \frac{d}{d x} [5^{\frac{1}{2}} x^{\frac{1}{2}}])$

Step 4.3

Replace all occurrences of $u_{2}$ with $5^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k (\sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) \frac{d}{d x} [5^{\frac{1}{2}} x^{\frac{1}{2}}])$

Step 5

Differentiate.

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Step 5.1

Since $5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $5^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 5.2

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

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 6

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

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 7

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

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 8

Combine the numerators over the common denominator.

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 9.2

Subtract $2$ from $1$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 10

Move the negative in front of the fraction.

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 11

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

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) (5^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2})$

Step 12

Combine $5^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) \frac{5^{\frac{1}{2}} x^{- \frac{1}{2}}}{2}$

Step 13

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

$e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 14

Combine $e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})}$ and $\frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}}) \frac{e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \cdot 5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 15

Combine $\frac{e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \cdot 5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ and $k$ .

$\frac{e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \cdot 5^{\frac{1}{2}} k}{2 x^{\frac{1}{2}}} \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})$

Step 16

Combine $\frac{e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \cdot 5^{\frac{1}{2}} k}{2 x^{\frac{1}{2}}}$ and $\sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})$ .

$\frac{e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \cdot 5^{\frac{1}{2}} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 17

Simplify.

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Step 17.1

Reorder factors in $e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \cdot 5^{\frac{1}{2}} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})$ .

$\frac{k e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \cdot 5^{\frac{1}{2}} \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 17.2

Reorder terms.

$\frac{5^{\frac{1}{2}} e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 17.3

Reorder factors in $\frac{5^{\frac{1}{2}} e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} k \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$ .

$\frac{k \cdot 5^{\frac{1}{2}} e^{k \tan (5^{\frac{1}{2}} x^{\frac{1}{2}})} \sec^{2} (5^{\frac{1}{2}} x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$