Calculus Examples

$y = \frac{5}{1 + e^{3 x}}$

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{1 + e^{3 x}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{1 + e^{3 x}}]$ .

$5 \frac{d}{d x} [\frac{1}{1 + e^{3 x}}]$

Step 1.2

Rewrite $\frac{1}{1 + e^{3 x}}$ as ${(1 + e^{3 x})}^{- 1}$ .

$5 \frac{d}{d x} [{(1 + e^{3 x})}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 1 + e^{3 x}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $1 + e^{3 x}$ .

$5 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [1 + e^{3 x}])$

Step 2.2

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

$5 (- {u_{1}}^{- 2} \frac{d}{d x} [1 + e^{3 x}])$

Step 2.3

Replace all occurrences of $u_{1}$ with $1 + e^{3 x}$ .

$5 (- {(1 + e^{3 x})}^{- 2} \frac{d}{d x} [1 + e^{3 x}])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $5$ .

$- 5 ({(1 + e^{3 x})}^{- 2} \frac{d}{d x} [1 + e^{3 x}])$

Step 3.2

By the Sum Rule, the derivative of $1 + e^{3 x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [e^{3 x}]$ .

$- 5 {(1 + e^{3 x})}^{- 2} (\frac{d}{d x} [1] + \frac{d}{d x} [e^{3 x}])$

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 5 {(1 + e^{3 x})}^{- 2} (0 + \frac{d}{d x} [e^{3 x}])$

Step 3.4

Add $0$ and $\frac{d}{d x} [e^{3 x}]$ .

$- 5 {(1 + e^{3 x})}^{- 2} \frac{d}{d x} [e^{3 x}]$

Step 4

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 $g (x) = 3 x$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$- 5 {(1 + e^{3 x})}^{- 2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [3 x])$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$- 5 {(1 + e^{3 x})}^{- 2} (e^{u_{2}} \frac{d}{d x} [3 x])$

Step 4.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$- 5 {(1 + e^{3 x})}^{- 2} (e^{3 x} \frac{d}{d x} [3 x])$

Step 5

Differentiate.

Tap for more steps...

Step 5.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- 5 {(1 + e^{3 x})}^{- 2} (e^{3 x} (3 \frac{d}{d x} [x]))$

Step 5.2

Multiply $3$ by $- 5$ .

$- 15 {(1 + e^{3 x})}^{- 2} (e^{3 x} (\frac{d}{d x} [x]))$

Step 5.3

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

$- 15 {(1 + e^{3 x})}^{- 2} (e^{3 x} \cdot 1)$

Step 5.4

Multiply $e^{3 x}$ by $1$ .

$- 15 {(1 + e^{3 x})}^{- 2} e^{3 x}$

Step 6

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

$- 15 \frac{1}{{(1 + e^{3 x})}^{2}} e^{3 x}$

Step 7

Combine terms.

Tap for more steps...

Step 7.1

Combine $- 15$ and $\frac{1}{{(1 + e^{3 x})}^{2}}$ .

$\frac{- 15}{{(1 + e^{3 x})}^{2}} e^{3 x}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{15}{{(1 + e^{3 x})}^{2}} e^{3 x}$

Step 7.3

Combine $e^{3 x}$ and $\frac{15}{{(1 + e^{3 x})}^{2}}$ .

$- \frac{e^{3 x} \cdot 15}{{(1 + e^{3 x})}^{2}}$

Step 7.4

Move $15$ to the left of $e^{3 x}$ .

$- \frac{15 e^{3 x}}{{(1 + e^{3 x})}^{2}}$