Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} + 1$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$x (2 x + 0) + (x^{2} + 1) \frac{d}{d x} [x]$

Step 2.4

Add $2 x$ and $0$ .

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

Step 3

Raise $x$ to the power of $1$ .

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

Step 4

Raise $x$ to the power of $1$ .

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

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6

Add $1$ and $1$ .

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

Step 7

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

$2 x^{2} + (x^{2} + 1) \cdot 1$

Step 8

Simplify by adding terms.

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Step 8.1

Multiply $x^{2} + 1$ by $1$ .

$2 x^{2} + x^{2} + 1$

Step 8.2

Add $2 x^{2}$ and $x^{2}$ .

$3 x^{2} + 1$