Calculus Examples

$y = \frac{1}{c + k e^{c}}$

Step 1

Rewrite $\frac{1}{c + k e^{c}}$ as ${(c + k e^{c})}^{- 1}$ .

$\frac{d}{d c} [{(c + k e^{c})}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d c} [f (g (c))]$ is $f' (g (c)) g' (c)$ where $f (c) = c^{- 1}$ and $g (c) = c + k e^{c}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $c + k e^{c}$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d c} [c + k e^{c}]$

Step 2.2

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

$- u^{- 2} \frac{d}{d c} [c + k e^{c}]$

Step 2.3

Replace all occurrences of $u$ with $c + k e^{c}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

$- {(c + k e^{c})}^{- 2} (1 + k e^{c})$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{{(c + k e^{c})}^{2}} (1 + k e^{c})$

Step 5.2

Reorder the factors of $- \frac{1}{{(c + k e^{c})}^{2}} (1 + k e^{c})$ .

$- (1 + k e^{c}) \frac{1}{{(c + k e^{c})}^{2}}$

Step 5.3

Apply the distributive property.

$(- 1 \cdot 1 - (k e^{c})) \frac{1}{{(c + k e^{c})}^{2}}$

Step 5.4

Multiply $- 1$ by $1$ .

$(- 1 - (k e^{c})) \frac{1}{{(c + k e^{c})}^{2}}$

Step 5.5

Multiply $- 1 - k e^{c}$ by $\frac{1}{{(c + k e^{c})}^{2}}$ .

$\frac{- 1 - k e^{c}}{{(c + k e^{c})}^{2}}$

Step 5.6

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) - k e^{c}}{{(c + k e^{c})}^{2}}$

Step 5.7

Factor $- 1$ out of $- k e^{c}$ .

$\frac{- 1 (1) - (k e^{c})}{{(c + k e^{c})}^{2}}$

Step 5.8

Factor $- 1$ out of $- 1 (1) - (k e^{c})$ .

$\frac{- 1 (1 + k e^{c})}{{(c + k e^{c})}^{2}}$

Step 5.9

Move the negative in front of the fraction.

$- \frac{1 + k e^{c}}{{(c + k e^{c})}^{2}}$