Calculus Examples

$y = \frac{5 x}{\log_{5} (x)}$

Step 1

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

$5 \frac{d}{d x} [\frac{x}{\log_{5} (x)}]$

Step 2

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 $f (x) = x$ and $g (x) = \log_{5} (x)$ .

$5 \frac{\log_{5} (x) \frac{d}{d x} [x] - x \frac{d}{d x} [\log_{5} (x)]}{{\log_{5}}^{2} (x)}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$5 \frac{\log_{5} (x) \cdot 1 - x \frac{d}{d x} [\log_{5} (x)]}{{\log_{5}}^{2} (x)}$

Step 3.2

Multiply $\log_{5} (x)$ by $1$ .

$5 \frac{\log_{5} (x) - x \frac{d}{d x} [\log_{5} (x)]}{{\log_{5}}^{2} (x)}$

Step 4

The derivative of $\log_{5} (x)$ with respect to $x$ is $\frac{1}{x \ln (5)}$ .

$5 \frac{\log_{5} (x) - x \frac{1}{x \ln (5)}}{{\log_{5}}^{2} (x)}$

Step 5

Simplify terms.

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Step 5.1

Combine $\frac{1}{x \ln (5)}$ and $x$ .

$5 \frac{\log_{5} (x) - \frac{x}{x \ln (5)}}{{\log_{5}}^{2} (x)}$

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

$5 \frac{\log_{5} (x) - \frac{x}{x \ln (5)}}{{\log_{5}}^{2} (x)}$

Step 5.2.2

Rewrite the expression.

$5 \frac{\log_{5} (x) - \frac{1}{\ln (5)}}{{\log_{5}}^{2} (x)}$

Step 5.3

Combine $5$ and $\frac{\log_{5} (x) - \frac{1}{\ln (5)}}{{\log_{5}}^{2} (x)}$ .

$\frac{5 (\log_{5} (x) - \frac{1}{\ln (5)})}{{\log_{5}}^{2} (x)}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{5 \log_{5} (x) + 5 (- \frac{1}{\ln (5)})}{{\log_{5}}^{2} (x)}$

Step 6.2

Simplify each term.

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Step 6.2.1

Simplify $5 \log_{5} (x)$ by moving $5$ inside the logarithm.

$\frac{\log_{5} (x^{5}) + 5 (- \frac{1}{\ln (5)})}{{\log_{5}}^{2} (x)}$

Step 6.2.2

Multiply $5 (- \frac{1}{\ln (5)})$ .

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Step 6.2.2.1

Multiply $- 1$ by $5$ .

$\frac{\log_{5} (x^{5}) - 5 \frac{1}{\ln (5)}}{{\log_{5}}^{2} (x)}$

Step 6.2.2.2

Combine $- 5$ and $\frac{1}{\ln (5)}$ .

$\frac{\log_{5} (x^{5}) + \frac{- 5}{\ln (5)}}{{\log_{5}}^{2} (x)}$

Step 6.2.3

Move the negative in front of the fraction.

$\frac{\log_{5} (x^{5}) - \frac{5}{\ln (5)}}{{\log_{5}}^{2} (x)}$