Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((x^{2} + 2) (\ln (x^{2} + 1)))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = x^{2} + 2$ and $g (x) = \ln (x^{2} + 1)$ .

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Combine fractions.

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Step 3.3.4.1

Add $2 x$ and $0$ .

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

Step 3.3.4.2

Combine $2$ and $\frac{1}{x^{2} + 1}$ .

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

Step 3.3.4.3

Combine $x$ and $\frac{2}{x^{2} + 1}$ .

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

Step 3.3.4.4

Move $2$ to the left of $x$ .

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$(x^{2} + 2) \frac{2 x}{x^{2} + 1} + \ln (x^{2} + 1) (2 x + 0)$

Step 3.3.8

Simplify the expression.

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Step 3.3.8.1

Add $2 x$ and $0$ .

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

Step 3.3.8.2

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = (x^{2} + 2) (\frac{2 x}{x^{2} + 1}) + 2 x \ln (x^{2} + 1)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = (x^{2} + 2) (\frac{2 x}{x^{2} + 1}) + 2 x \ln (x^{2} + 1)$