Calculus Examples

$f (x) = 6 \log_{2} (x)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \log_{2} (x)$ with respect to $x$ is $6 \frac{d}{d x} [\log_{2} (x)]$ .

$6 \frac{d}{d x} [\log_{2} (x)]$

Step 1.2

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

$6 \frac{1}{x \ln (2)}$

Step 1.3

Combine $6$ and $\frac{1}{x \ln (2)}$ .

$f' (x) = \frac{6}{x \ln (2)}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $\frac{6}{\ln (2)}$ is constant with respect to $x$ , the derivative of $\frac{6}{x \ln (2)}$ with respect to $x$ is $\frac{6}{\ln (2)} \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{6}{\ln (2)} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.3

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

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

Step 2.4

Combine fractions.

Tap for more steps...

Step 2.4.1

Combine $\frac{6}{\ln (2)}$ and $x^{- 2}$ .

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

Step 2.4.2

Simplify the expression.

Tap for more steps...

Step 2.4.2.1

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.2.2

Reorder factors in $- \frac{6}{\ln (2) x^{2}}$ .

$f'' (x) = - \frac{6}{x^{2} \ln (2)}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{6}{x^{2} \ln (2)}$ .

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