Algebra Examples

$y = \frac{e^{6 x}}{\ln (4 x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{6 x}$ and $g (x) = \ln (4 x)$ .

$\frac{\ln (4 x) \frac{d}{d x} [e^{6 x}] - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 6 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $6 x$ .

$\frac{\ln (4 x) (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

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 $a$ = $e$ .

$\frac{\ln (4 x) (e^{u_{1}} \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

Step 2.3

Replace all occurrences of $u_{1}$ with $6 x$ .

$\frac{\ln (4 x) (e^{6 x} \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

Step 3

Differentiate.

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Step 3.1

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

$\frac{\ln (4 x) (e^{6 x} (6 \frac{d}{d x} [x])) - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

Step 3.2

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

$\frac{\ln (4 x) (e^{6 x} (6 \cdot 1)) - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $6$ by $1$ .

$\frac{\ln (4 x) (e^{6 x} \cdot 6) - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

Step 3.3.2

Move $6$ to the left of $e^{6 x}$ .

$\frac{\ln (4 x) (6 \cdot e^{6 x}) - e^{6 x} \frac{d}{d x} [\ln (4 x)]}{\ln^{2} (4 x)}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 4 x$ .

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Step 4.1

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

$\frac{\ln (4 x) (6 e^{6 x}) - e^{6 x} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [4 x])}{\ln^{2} (4 x)}$

Step 4.2

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

$\frac{\ln (4 x) (6 e^{6 x}) - e^{6 x} (\frac{1}{u_{2}} \frac{d}{d x} [4 x])}{\ln^{2} (4 x)}$

Step 4.3

Replace all occurrences of $u_{2}$ with $4 x$ .

$\frac{\ln (4 x) (6 e^{6 x}) - e^{6 x} (\frac{1}{4 x} \frac{d}{d x} [4 x])}{\ln^{2} (4 x)}$

Step 5

Differentiate.

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Step 5.1

Combine $\frac{1}{4 x}$ and $e^{6 x}$ .

$\frac{\ln (4 x) (6 e^{6 x}) - \frac{e^{6 x}}{4 x} \frac{d}{d x} [4 x]}{\ln^{2} (4 x)}$

Step 5.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{\ln (4 x) (6 e^{6 x}) - \frac{e^{6 x}}{4 x} (4 \frac{d}{d x} [x])}{\ln^{2} (4 x)}$

Step 5.3

Simplify terms.

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Step 5.3.1

Multiply $4$ by $- 1$ .

$\frac{\ln (4 x) (6 e^{6 x}) - 4 \frac{e^{6 x}}{4 x} \frac{d}{d x} [x]}{\ln^{2} (4 x)}$

Step 5.3.2

Combine $- 4$ and $\frac{e^{6 x}}{4 x}$ .

$\frac{\ln (4 x) (6 e^{6 x}) + \frac{- 4 e^{6 x}}{4 x} \frac{d}{d x} [x]}{\ln^{2} (4 x)}$

Step 5.3.3

Cancel the common factor of $- 4$ and $4$ .

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Step 5.3.3.1

Factor $4$ out of $- 4 e^{6 x}$ .

$\frac{\ln (4 x) (6 e^{6 x}) + \frac{4 (- e^{6 x})}{4 x} \frac{d}{d x} [x]}{\ln^{2} (4 x)}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $4$ out of $4 x$ .

$\frac{\ln (4 x) (6 e^{6 x}) + \frac{4 (- e^{6 x})}{4 (x)} \frac{d}{d x} [x]}{\ln^{2} (4 x)}$

Step 5.3.3.2.2

Cancel the common factor.

$\frac{\ln (4 x) (6 e^{6 x}) + \frac{4 (- e^{6 x})}{4 x} \frac{d}{d x} [x]}{\ln^{2} (4 x)}$

Step 5.3.3.2.3

Rewrite the expression.

$\frac{\ln (4 x) (6 e^{6 x}) + \frac{- e^{6 x}}{x} \frac{d}{d x} [x]}{\ln^{2} (4 x)}$

Step 5.3.4

Move the negative in front of the fraction.

$\frac{\ln (4 x) (6 e^{6 x}) - \frac{e^{6 x}}{x} \frac{d}{d x} [x]}{\ln^{2} (4 x)}$

Step 5.4

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

$\frac{\ln (4 x) (6 e^{6 x}) - \frac{e^{6 x}}{x} \cdot 1}{\ln^{2} (4 x)}$

Step 5.5

Multiply $- 1$ by $1$ .

$\frac{\ln (4 x) (6 e^{6 x}) - \frac{e^{6 x}}{x}}{\ln^{2} (4 x)}$

Step 6

Simplify the numerator.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite using the commutative property of multiplication.

$\frac{6 \ln (4 x) e^{6 x} - \frac{e^{6 x}}{x}}{\ln^{2} (4 x)}$

Step 6.1.2

Simplify $6 \ln (4 x)$ by moving $6$ inside the logarithm.

$\frac{\ln ({(4 x)}^{6}) e^{6 x} - \frac{e^{6 x}}{x}}{\ln^{2} (4 x)}$

Step 6.1.3

Apply the product rule to $4 x$ .

$\frac{\ln (4^{6} x^{6}) e^{6 x} - \frac{e^{6 x}}{x}}{\ln^{2} (4 x)}$

Step 6.1.4

Raise $4$ to the power of $6$ .

$\frac{\ln (4096 x^{6}) e^{6 x} - \frac{e^{6 x}}{x}}{\ln^{2} (4 x)}$

Step 6.2

Reorder factors in $\ln (4096 x^{6}) e^{6 x} - \frac{e^{6 x}}{x}$ .

$\frac{e^{6 x} \ln (4096 x^{6}) - \frac{e^{6 x}}{x}}{\ln^{2} (4 x)}$