Calculus Examples

$f (x) = e^{\log (e^{5 x})}$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\log (e^{5 x})$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\log (e^{5 x})$ .

$e^{\log (e^{5 x})} \frac{d}{d x} [\log (e^{5 x})]$

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $\log (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2} \ln (10)}$ .

$e^{\log (e^{5 x})} (\frac{1}{u_{2} \ln (10)} \frac{d}{d x} [e^{5 x}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $e^{5 x}$ .

$e^{\log (e^{5 x})} (\frac{1}{e^{5 x} \ln (10)} \frac{d}{d x} [e^{5 x}])$

Step 3

Simplify terms.

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Step 3.1

Combine $\frac{1}{e^{5 x} \ln (10)}$ and $e^{\log (e^{5 x})}$ .

$\frac{e^{\log (e^{5 x})}}{e^{5 x} \ln (10)} \frac{d}{d x} [e^{5 x}]$

Step 3.2

Cancel the common factor of $e^{\log (e^{5 x})}$ and $e^{5 x}$ .

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Step 3.2.1

Factor $e^{5 x}$ out of $e^{\log (e^{5 x})}$ .

$\frac{e^{5 x} e^{\log (e^{5 x}) - 5 x}}{e^{5 x} \ln (10)} \frac{d}{d x} [e^{5 x}]$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $e^{5 x}$ out of $e^{5 x} \ln (10)$ .

$\frac{e^{5 x} e^{\log (e^{5 x}) - 5 x}}{e^{5 x} (\ln (10))} \frac{d}{d x} [e^{5 x}]$

Step 3.2.2.2

Cancel the common factor.

$\frac{e^{5 x} e^{\log (e^{5 x}) - 5 x}}{e^{5 x} \ln (10)} \frac{d}{d x} [e^{5 x}]$

Step 3.2.2.3

Rewrite the expression.

$\frac{e^{\log (e^{5 x}) - 5 x}}{\ln (10)} \frac{d}{d x} [e^{5 x}]$

Step 4

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

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Step 4.1

To apply the Chain Rule, set $u_{3}$ as $5 x$ .

$\frac{e^{\log (e^{5 x}) - 5 x}}{\ln (10)} (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [5 x])$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$\frac{e^{\log (e^{5 x}) - 5 x}}{\ln (10)} (e^{u_{3}} \frac{d}{d x} [5 x])$

Step 4.3

Replace all occurrences of $u_{3}$ with $5 x$ .

$\frac{e^{\log (e^{5 x}) - 5 x}}{\ln (10)} (e^{5 x} \frac{d}{d x} [5 x])$

Step 5

Combine $e^{5 x}$ and $\frac{e^{\log (e^{5 x}) - 5 x}}{\ln (10)}$ .

$\frac{e^{5 x} e^{\log (e^{5 x}) - 5 x}}{\ln (10)} \frac{d}{d x} [5 x]$

Step 6

Multiply $e^{5 x}$ by $e^{\log (e^{5 x}) - 5 x}$ by adding the exponents.

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Step 6.1

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

$\frac{e^{5 x + \log (e^{5 x}) - 5 x}}{\ln (10)} \frac{d}{d x} [5 x]$

Step 6.2

Combine the opposite terms in $5 x + \log (e^{5 x}) - 5 x$ .

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Step 6.2.1

Subtract $5 x$ from $5 x$ .

$\frac{e^{0 + \log (e^{5 x})}}{\ln (10)} \frac{d}{d x} [5 x]$

Step 6.2.2

Add $0$ and $\log (e^{5 x})$ .

$\frac{e^{\log (e^{5 x})}}{\ln (10)} \frac{d}{d x} [5 x]$

Step 7

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

$\frac{e^{\log (e^{5 x})}}{\ln (10)} (5 \frac{d}{d x} [x])$

Step 8

Combine $5$ and $\frac{e^{\log (e^{5 x})}}{\ln (10)}$ .

$\frac{5 e^{\log (e^{5 x})}}{\ln (10)} \frac{d}{d x} [x]$

Step 9

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

$\frac{5 e^{\log (e^{5 x})}}{\ln (10)} \cdot 1$

Step 10

Multiply $\frac{5 e^{\log (e^{5 x})}}{\ln (10)}$ by $1$ .

$\frac{5 e^{\log (e^{5 x})}}{\ln (10)}$