Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- 1$ by $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

Move the negative in front of the fraction.

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