Calculus Examples

$y = \frac{7^{x}}{\log_{7} (x)}$

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 $f (x) = 7^{x}$ and $g (x) = \log_{7} (x)$ .

$\frac{\log_{7} (x) \frac{d}{d x} [7^{x}] - 7^{x} \frac{d}{d x} [\log_{7} (x)]}{{\log_{7}}^{2} (x)}$

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $7$ .

$\frac{\log_{7} (x) (7^{x} \ln (7)) - 7^{x} \frac{d}{d x} [\log_{7} (x)]}{{\log_{7}}^{2} (x)}$

Step 3

The derivative of $\log_{7} (x)$ with respect to $x$ is $\frac{1}{x \ln (7)}$ .

$\frac{\log_{7} (x) (7^{x} \ln (7)) - 7^{x} \frac{1}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 4

Combine $\frac{1}{x \ln (7)}$ and $7^{x}$ .

$\frac{\log_{7} (x) (7^{x} \ln (7)) - \frac{7^{x}}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

$\frac{\ln (7) \log_{7} (x) \cdot 7^{x} - \frac{7^{x}}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.1.2

Reorder factors in $\ln (7) \log_{7} (x) \cdot 7^{x} - \frac{7^{x}}{x \ln (7)}$ .

$\frac{7^{x} \ln (7) \log_{7} (x) - \frac{7^{x}}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Factor $7^{x}$ out of $7^{x} \ln (7) \log_{7} (x) - \frac{7^{x}}{x \ln (7)}$ .

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Step 5.2.1.1

Factor $7^{x}$ out of $7^{x} \ln (7) \log_{7} (x)$ .

$\frac{7^{x} (\ln (7) \log_{7} (x)) - \frac{7^{x}}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.2.1.2

Factor $7^{x}$ out of $- \frac{7^{x}}{x \ln (7)}$ .

$\frac{7^{x} (\ln (7) \log_{7} (x)) + 7^{x} (- \frac{1}{x \ln (7)})}{{\log_{7}}^{2} (x)}$

Step 5.2.1.3

Factor $7^{x}$ out of $7^{x} (\ln (7) \log_{7} (x)) + 7^{x} (- \frac{1}{x \ln (7)})$ .

$\frac{7^{x} (\ln (7) \log_{7} (x) - \frac{1}{x \ln (7)})}{{\log_{7}}^{2} (x)}$

Step 5.2.2

To write $\ln (7) \log_{7} (x)$ as a fraction with a common denominator, multiply by $\frac{x \ln (7)}{x \ln (7)}$ .

$\frac{7^{x} (\ln (7) \log_{7} (x) \cdot \frac{x \ln (7)}{x \ln (7)} - \frac{1}{x \ln (7)})}{{\log_{7}}^{2} (x)}$

Step 5.2.3

Combine $\ln (7) \log_{7} (x)$ and $\frac{x \ln (7)}{x \ln (7)}$ .

$\frac{7^{x} (\frac{\ln (7) \log_{7} (x) (x \ln (7))}{x \ln (7)} - \frac{1}{x \ln (7)})}{{\log_{7}}^{2} (x)}$

Step 5.2.4

Combine the numerators over the common denominator.

$\frac{7^{x} \frac{\ln (7) \log_{7} (x) (x \ln (7)) - 1}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $\ln (7) \log_{7} (x) (x \ln (7))$ .

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Step 5.2.5.1.1

Raise $\ln (7)$ to the power of $1$ .

$\frac{7^{x} \frac{\log_{7} (x) (x (\ln^{1} (7) \ln (7))) - 1}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.2.5.1.2

Raise $\ln (7)$ to the power of $1$ .

$\frac{7^{x} \frac{\log_{7} (x) (x (\ln^{1} (7) \ln^{1} (7))) - 1}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.2.5.1.3

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

$\frac{7^{x} \frac{\log_{7} (x) (x {\ln (7)}^{1 + 1}) - 1}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.2.5.1.4

Add $1$ and $1$ .

$\frac{7^{x} \frac{\log_{7} (x) (x \ln^{2} (7)) - 1}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.2.5.2

Rewrite using the commutative property of multiplication.

$\frac{7^{x} \frac{\ln^{2} (7) \log_{7} (x) x - 1}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.3

Combine $7^{x}$ and $\frac{\ln^{2} (7) \log_{7} (x) x - 1}{x \ln (7)}$ .

$\frac{\frac{7^{x} (\ln^{2} (7) \log_{7} (x) x - 1)}{x \ln (7)}}{{\log_{7}}^{2} (x)}$

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{7^{x} (\ln^{2} (7) \log_{7} (x) x - 1)}{x \ln (7)} \cdot \frac{1}{{\log_{7}}^{2} (x)}$

Step 5.5

Combine.

$\frac{7^{x} (\ln^{2} (7) \log_{7} (x) x - 1) \cdot 1}{x \ln (7) {\log_{7}}^{2} (x)}$

Step 5.6

Multiply $7^{x}$ by $1$ .

$\frac{7^{x} (\ln^{2} (7) \log_{7} (x) x - 1)}{x \ln (7) {\log_{7}}^{2} (x)}$

Step 5.7

Reorder factors in $\frac{7^{x} (\ln^{2} (7) \log_{7} (x) x - 1)}{x \ln (7) {\log_{7}}^{2} (x)}$ .

$\frac{7^{x} (x \ln^{2} (7) \log_{7} (x) - 1)}{x \ln (7) {\log_{7}}^{2} (x)}$