Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

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

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $- 2 x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $- 2 x$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 3.3.2

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

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

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Multiply $- 2$ by $1$ .

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

Step 3.3.3.2

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

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

Step 3.3.4

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

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

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

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

Step 3.4.2

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

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

Step 4

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

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

Step 5

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

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