Calculus Examples

$y = \frac{6}{(1 + e^{- x})}$ , $(0, 3)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 3$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{1 + e^{- x}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{1 + e^{- x}}]$ .

$6 \frac{d}{d x} [\frac{1}{1 + e^{- x}}]$

Step 1.1.2

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

$6 \frac{d}{d x} [{(1 + e^{- x})}^{- 1}]$

Step 1.2

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 $g (x) = 1 + e^{- x}$ .

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Step 1.2.1

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

$6 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [1 + e^{- x}])$

Step 1.2.2

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

$6 (- {u_{1}}^{- 2} \frac{d}{d x} [1 + e^{- x}])$

Step 1.2.3

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

$6 (- {(1 + e^{- x})}^{- 2} \frac{d}{d x} [1 + e^{- x}])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $6$ .

$- 6 ({(1 + e^{- x})}^{- 2} \frac{d}{d x} [1 + e^{- x}])$

Step 1.3.2

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

$- 6 {(1 + e^{- x})}^{- 2} (\frac{d}{d x} [1] + \frac{d}{d x} [e^{- x}])$

Step 1.3.3

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

$- 6 {(1 + e^{- x})}^{- 2} (0 + \frac{d}{d x} [e^{- x}])$

Step 1.3.4

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

$- 6 {(1 + e^{- x})}^{- 2} \frac{d}{d x} [e^{- x}]$

Step 1.4

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 $g (x) = - x$ .

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Step 1.4.1

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

$- 6 {(1 + e^{- x})}^{- 2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [- x])$

Step 1.4.2

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

$- 6 {(1 + e^{- x})}^{- 2} (e^{u_{2}} \frac{d}{d x} [- x])$

Step 1.4.3

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

$- 6 {(1 + e^{- x})}^{- 2} (e^{- x} \frac{d}{d x} [- x])$

Step 1.5

Differentiate.

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Step 1.5.1

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

$- 6 {(1 + e^{- x})}^{- 2} (e^{- x} (- \frac{d}{d x} [x]))$

Step 1.5.2

Multiply $- 1$ by $- 6$ .

$6 {(1 + e^{- x})}^{- 2} (e^{- x} (\frac{d}{d x} [x]))$

Step 1.5.3

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

$6 {(1 + e^{- x})}^{- 2} (e^{- x} \cdot 1)$

Step 1.5.4

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

$6 {(1 + e^{- x})}^{- 2} e^{- x}$

Step 1.6

Simplify.

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Step 1.6.1

Reorder the factors of $6 {(1 + e^{- x})}^{- 2} e^{- x}$ .

$6 e^{- x} {(1 + e^{- x})}^{- 2}$

Step 1.6.2

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

$6 e^{- x} \frac{1}{{(1 + e^{- x})}^{2}}$

Step 1.6.3

Multiply $6 e^{- x} \frac{1}{{(1 + e^{- x})}^{2}}$ .

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Step 1.6.3.1

Combine $\frac{1}{{(1 + e^{- x})}^{2}}$ and $6$ .

$\frac{6}{{(1 + e^{- x})}^{2}} e^{- x}$

Step 1.6.3.2

Combine $\frac{6}{{(1 + e^{- x})}^{2}}$ and $e^{- x}$ .

$\frac{6 e^{- x}}{{(1 + e^{- x})}^{2}}$

Step 1.7

Evaluate the derivative at $x = 0$ .

$\frac{6 e^{- (0)}}{{(1 + e^{- (0)})}^{2}}$

Step 1.8

Simplify.

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Step 1.8.1

Simplify the numerator.

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Step 1.8.1.1

Multiply $- 1$ by $0$ .

$\frac{6 e^{0}}{{(1 + e^{- (0)})}^{2}}$

Step 1.8.1.2

Anything raised to $0$ is $1$ .

$\frac{6 \cdot 1}{{(1 + e^{- (0)})}^{2}}$

Step 1.8.2

Simplify the denominator.

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Step 1.8.2.1

Multiply $- 1$ by $0$ .

$\frac{6 \cdot 1}{{(1 + e^{0})}^{2}}$

Step 1.8.2.2

Anything raised to $0$ is $1$ .

$\frac{6 \cdot 1}{{(1 + 1)}^{2}}$

Step 1.8.2.3

Add $1$ and $1$ .

$\frac{6 \cdot 1}{2^{2}}$

Step 1.8.2.4

Raise $2$ to the power of $2$ .

$\frac{6 \cdot 1}{4}$

Step 1.8.3

Reduce the expression by cancelling the common factors.

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Step 1.8.3.1

Multiply $6$ by $1$ .

$\frac{6}{4}$

Step 1.8.3.2

Cancel the common factor of $6$ and $4$ .

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Step 1.8.3.2.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4}$

Step 1.8.3.2.2

Cancel the common factors.

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Step 1.8.3.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 3}{2 \cdot 2}$

Step 1.8.3.2.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 2}$

Step 1.8.3.2.2.3

Rewrite the expression.

$\frac{3}{2}$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $\frac{3}{2}$ and a given point $(0, 3)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = \frac{3}{2} \cdot (x - (0))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 3 = \frac{3}{2} \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{3}{2} \cdot (x + 0)$ .

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Step 2.3.1.1

Add $x$ and $0$ .

$y - 3 = \frac{3}{2} \cdot x$

Step 2.3.1.2

Combine $\frac{3}{2}$ and $x$ .

$y - 3 = \frac{3 x}{2}$

Step 2.3.2

Add $3$ to both sides of the equation.

$y = \frac{3 x}{2} + 3$

Step 2.3.3

Reorder terms.

$y = \frac{3}{2} x + 3$

Step 3