Calculus Examples

$y = x^{2} (e^{x} - 1)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} (e^{x} - 1))$

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 $f (x) = x^{2}$ and $g (x) = e^{x} - 1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

Add $e^{x}$ and $0$ .

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

Step 3.4.3

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

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

Step 3.4.4

Move $2$ to the left of $e^{x} - 1$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

Multiply $2$ by $- 1$ .

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

Step 3.5.4

Reorder terms.

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

Step 3.5.5

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

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

Step 4

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

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

Step 5

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

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