Calculus Examples

$x^{2} \ln (x) - 3 x e^{x}$

Step 1

By the Sum Rule, the derivative of $x^{2} \ln (x) - 3 x e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} \ln (x)] + \frac{d}{d x} [- 3 x e^{x}]$ .

$\frac{d}{d x} [x^{2} \ln (x)] + \frac{d}{d x} [- 3 x e^{x}]$

Step 2

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

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Step 2.1

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

$x^{2} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.2

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

$x^{2} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.3

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

$x^{2} \frac{1}{x} + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.4

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

$\frac{x^{2}}{x} + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.5

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.5.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x}{x} + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

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

$\frac{x \cdot x}{x^{1}} + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.5.2.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.5.2.3

Cancel the common factor.

$\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.5.2.4

Rewrite the expression.

$\frac{x}{1} + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 2.5.2.5

Divide $x$ by $1$ .

$x + \ln (x) (2 x) + \frac{d}{d x} [- 3 x e^{x}]$

Step 3

Evaluate $\frac{d}{d x} [- 3 x e^{x}]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x e^{x}$ with respect to $x$ is $- 3 \frac{d}{d x} [x e^{x}]$ .

$x + \ln (x) (2 x) - 3 \frac{d}{d x} [x e^{x}]$

Step 3.2

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 $g (x) = e^{x}$ .

$x + \ln (x) (2 x) - 3 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x])$

Step 3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$x + \ln (x) (2 x) - 3 (x e^{x} + e^{x} \frac{d}{d x} [x])$

Step 3.4

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

$x + \ln (x) (2 x) - 3 (x e^{x} + e^{x} \cdot 1)$

Step 3.5

Multiply $e^{x}$ by $1$ .

$x + \ln (x) (2 x) - 3 (x e^{x} + e^{x})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$x + \ln (x) (2 x) - 3 (x e^{x}) - 3 e^{x}$

Step 4.2

Remove unnecessary parentheses.

$x + \ln (x) (2 x) - 3 x e^{x} - 3 e^{x}$

Step 4.3

Reorder terms.

$2 x \ln (x) - 3 e^{x} x + x - 3 e^{x}$

Step 4.4

Reorder factors in $2 x \ln (x) - 3 e^{x} x + x - 3 e^{x}$ .

$2 x \ln (x) - 3 x e^{x} + x - 3 e^{x}$