Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \frac{x}{2}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $- \frac{x}{2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $- \frac{x}{2}$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{x}{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $- 1$ by $1$ .

$- \frac{e^{- \frac{x}{2}}}{2}$

Step 4

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

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

Step 5

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

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