Calculus Examples

$f (x) = \log (4) (x^{5} + 1)$

Step 1

Since $\log (4)$ is constant with respect to $x$ , the derivative of $\log (4) (x^{5} + 1)$ with respect to $x$ is $\log (4) \frac{d}{d x} [x^{5} + 1]$ .

$\log (4) \frac{d}{d x} [x^{5} + 1]$

Step 2

By the Sum Rule, the derivative of $x^{5} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [1]$ .

$\log (4) (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [1])$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\log (4) (5 x^{4} + \frac{d}{d x} [1])$

Step 4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\log (4) (5 x^{4} + 0)$

Step 5

Simplify the expression.

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Step 5.1

Add $5 x^{4}$ and $0$ .

$\log (4) (5 x^{4})$

Step 5.2

Move $5$ to the left of $\log (4)$ .

$5 \cdot \log (4) x^{4}$

Step 5.3

Reorder the factors of $5 \log (4) x^{4}$ .

$5 x^{4} \log (4)$