Calculus Examples

$f (x) = e^{x^{3} - 1}$ at $x = 1$

Step 1

Find the corresponding $y$ -value to $x = 1$ .

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Step 1.1

Substitute $1$ in for $x$ .

$y = e^{{(1)}^{3} - 1}$

Step 1.2

Simplify $e^{{(1)}^{3} - 1}$ .

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Step 1.2.1

One to any power is one.

$y = e^{1 - 1}$

Step 1.2.2

Subtract $1$ from $1$ .

$y = e^{0}$

Step 1.2.3

Anything raised to $0$ is $1$ .

$y = 1$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 1$ to find the slope of the tangent line.

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Step 2.1

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 $g (x) = x^{3} - 1$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x^{3} - 1$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{3} - 1]$

Step 2.1.2

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

$e^{u} \frac{d}{d x} [x^{3} - 1]$

Step 2.1.3

Replace all occurrences of $u$ with $x^{3} - 1$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.2.3

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

$e^{x^{3} - 1} (3 x^{2} + 0)$

Step 2.2.4

Add $3 x^{2}$ and $0$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Reorder the factors of $e^{x^{3} - 1} (3 x^{2})$ .

$3 e^{x^{3} - 1} x^{2}$

Step 2.3.2

Reorder factors in $3 e^{x^{3} - 1} x^{2}$ .

$3 x^{2} e^{x^{3} - 1}$

Step 2.4

Evaluate the derivative at $x = 1$ .

$3 {(1)}^{2} e^{{(1)}^{3} - 1}$

Step 2.5

Simplify.

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Step 2.5.1

One to any power is one.

$3 \cdot 1 e^{{(1)}^{3} - 1}$

Step 2.5.2

Multiply $3$ by $1$ .

$3 e^{{(1)}^{3} - 1}$

Step 2.5.3

One to any power is one.

$3 e^{1 - 1}$

Step 2.5.4

Subtract $1$ from $1$ .

$3 e^{0}$

Step 2.5.5

Anything raised to $0$ is $1$ .

$3 \cdot 1$

Step 2.5.6

Multiply $3$ by $1$ .

$3$

Step 3

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

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Step 3.1

Use the slope $3$ and a given point $(1, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 3 \cdot (x - (1))$

Step 3.2

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

$y - 1 = 3 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $3 \cdot (x - 1)$ .

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Step 3.3.1.1

Rewrite.

$y - 1 = 0 + 0 + 3 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 1 = 3 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y - 1 = 3 x + 3 \cdot - 1$

Step 3.3.1.4

Multiply $3$ by $- 1$ .

$y - 1 = 3 x - 3$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $1$ to both sides of the equation.

$y = 3 x - 3 + 1$

Step 3.3.2.2

Add $- 3$ and $1$ .

$y = 3 x - 2$

Step 4