Calculus Examples

$y = \log (x) + 4 x^{2} + 1$

Step 1

By the Sum Rule, the derivative of $\log (x) + 4 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [\log (x)] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [\log (x)] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [1]$

Step 2

The derivative of $\log (x)$ with respect to $x$ is $\frac{1}{x \ln (10)}$ .

$\frac{1}{x \ln (10)} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [1]$

Step 3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{x \ln (10)} + 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{x \ln (10)} + 4 (2 x) + \frac{d}{d x} [1]$

Step 3.3

Multiply $2$ by $4$ .

$\frac{1}{x \ln (10)} + 8 x + \frac{d}{d x} [1]$

Step 4

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

$\frac{1}{x \ln (10)} + 8 x + 0$

Step 5

Simplify.

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Step 5.1

Add $\frac{1}{x \ln (10)} + 8 x$ and $0$ .

$\frac{1}{x \ln (10)} + 8 x$

Step 5.2

Reorder terms.

$8 x + \frac{1}{x \ln (10)}$