Calculus Examples

x^{2} \ln (x)

$x^{2} \ln (x)$

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x^{2}

$f (x) = x^{2}$ and

g (x) = \ln (x)

$g (x) = \ln (x)$ .

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

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

Step 2

The derivative of

\ln (x)

$\ln (x)$ with respect to

x

$x$ is

\frac{1}{x}

$\frac{1}{x}$ .

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

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

Step 3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.1

Combine

x^{2}

$x^{2}$ and

\frac{1}{x}

$\frac{1}{x}$ .

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

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

Step 3.2

Cancel the common factor of

x^{2}

$x^{2}$ and

x

$x$ .

Tap for more steps...

Step 3.2.1

Factor

x

$x$ out of

x^{2}

$x^{2}$ .

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

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

Step 3.2.2

Cancel the common factors.

Tap for more steps...

Step 3.2.2.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 3.2.2.2

Factor

x

$x$ out of

x^{1}

$x^{1}$ .

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

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

Step 3.2.2.3

Cancel the common factor.

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

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

Step 3.2.2.4

Rewrite the expression.

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

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

Step 3.2.2.5

Divide

x

$x$ by

1

$1$ .

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

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

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

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

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

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

Step 3.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

x + \ln (x) (2 x)

$x + \ln (x) (2 x)$

Step 3.4

Reorder terms.

2 x \ln (x) + x

$2 x \ln (x) + x$

2 x \ln (x) + x

$2 x \ln (x) + x$