Calculus Examples

$\frac{1}{e^{y}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\frac{1}{e^{y}}$ as ${(e^{y})}^{- 1}$ .

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

Step 1.2

Multiply the exponents in ${(e^{y})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d y} [e^{y \cdot - 1}]$

Step 1.2.2

Move $- 1$ to the left of $y$ .

$\frac{d}{d y} [e^{- 1 \cdot y}]$

Step 1.2.3

Rewrite $- 1 y$ as $- y$ .

$\frac{d}{d y} [e^{- y}]$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $- y$ .

$e^{- y} \frac{d}{d y} [- y]$

Step 3

Differentiate.

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Step 3.1

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

$e^{- y} (- \frac{d}{d y} [y])$

Step 3.2

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

$e^{- y} (- 1 \cdot 1)$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $- 1$ by $1$ .

$e^{- y} \cdot - 1$

Step 3.3.2

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

$- 1 \cdot e^{- y}$

Step 3.3.3

Rewrite $- 1 e^{- y}$ as $- e^{- y}$ .

$- e^{- y}$