Calculus Examples

$y = - \log_{2} (x) + 10 \ln (x)$

Step 1

By the Sum Rule, the derivative of $- \log_{2} (x) + 10 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [- \log_{2} (x)] + \frac{d}{d x} [10 \ln (x)]$ .

$\frac{d}{d x} [- \log_{2} (x)] + \frac{d}{d x} [10 \ln (x)]$

Step 2

Evaluate $\frac{d}{d x} [- \log_{2} (x)]$ .

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Step 2.1

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

$- \frac{d}{d x} [\log_{2} (x)] + \frac{d}{d x} [10 \ln (x)]$

Step 2.2

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

$- \frac{1}{x \ln (2)} + \frac{d}{d x} [10 \ln (x)]$

Step 3

Evaluate $\frac{d}{d x} [10 \ln (x)]$ .

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Step 3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 \ln (x)$ with respect to $x$ is $10 \frac{d}{d x} [\ln (x)]$ .

$- \frac{1}{x \ln (2)} + 10 \frac{d}{d x} [\ln (x)]$

Step 3.2

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

$- \frac{1}{x \ln (2)} + 10 \frac{1}{x}$

Step 3.3

Combine $10$ and $\frac{1}{x}$ .

$- \frac{1}{x \ln (2)} + \frac{10}{x}$

Step 4

Reorder terms.

$\frac{10}{x} - \frac{1}{x \ln (2)}$