Calculus Examples

$y = x^{2} e^{x} - x e^{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} e^{x} - x e^{x})$

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

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

$\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [- x e^{x}]$

Step 3.2

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

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Step 3.2.1

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

$x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x e^{x}]$

Step 3.2.2

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

$x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x e^{x}]$

Step 3.2.3

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

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

Step 3.3

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

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Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x e^{x}$ with respect to $x$ is $- \frac{d}{d x} [x e^{x}]$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

$x^{2} e^{x} + e^{x} (2 x) - (x e^{x} + e^{x})$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$x^{2} e^{x} + e^{x} (2 x) - (x e^{x}) - e^{x}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Subtract $x e^{x}$ from $e^{x} (2 x)$ .

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Step 3.4.2.1.1

Move $e^{x}$ .

$x^{2} e^{x} + 2 x e^{x} - x e^{x} - e^{x}$

Step 3.4.2.1.2

Subtract $x e^{x}$ from $2 x e^{x}$ .

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

Step 3.4.2.2

Multiply $x$ by $1$ .

$x^{2} e^{x} + x e^{x} - e^{x}$

Step 3.4.3

Reorder terms.

$e^{x} x^{2} + e^{x} x - e^{x}$

Step 3.4.4

Reorder factors in $e^{x} x^{2} + e^{x} x - e^{x}$ .

$x^{2} e^{x} + x e^{x} - e^{x}$

Step 4

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

$y' = x^{2} e^{x} + x e^{x} - e^{x}$

Step 5

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

$\frac{d y}{d x} = x^{2} e^{x} + x e^{x} - e^{x}$