Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

$f' (x) = \frac{d}{d x} (e^{x}) + \frac{d}{d x} (- x^{3})$

Step 1.1.2

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

$f' (x) = e^{x} + \frac{d}{d x} (- x^{3})$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x^{3}]$ .

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Step 1.1.3.1

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

$f' (x) = e^{x} - \frac{d}{d x} x^{3}$

Step 1.1.3.2

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

$f' (x) = e^{x} - (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $- 1$ .

$f' (x) = e^{x} - 3 x^{2}$

Step 1.1.4

Reorder terms.

$f' (x) = - 3 x^{2} + e^{x}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{2} + e^{x}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $- 3 x^{2} + e^{x} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 3 x^{2} + e^{x} = 0$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.45896226, 0.91000757, 3.73307902$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $e^{x} - x^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 0.45896226$ .

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Step 4.1.1

Substitute $- 0.45896226$ for $x$ .

$e^{- 0.45896226} - {(- 0.45896226)}^{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$\frac{1}{e^{0.45896226}} - {(- 0.45896226)}^{3}$

Step 4.1.2.1.2

Raise $- 0.45896226$ to the power of $3$ .

$\frac{1}{e^{0.45896226}} - - 0.09667873$

Step 4.1.2.1.3

Multiply $- 1$ by $- 0.09667873$ .

$\frac{1}{e^{0.45896226}} + 0.09667873$

Step 4.1.2.2

Add $\frac{1}{e^{0.45896226}}$ and $0.09667873$ .

$0.72861782$

Step 4.2

Evaluate at $x = 0.91000757$ .

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Step 4.2.1

Substitute $0.91000757$ for $x$ .

$e^{0.91000757} - {(0.91000757)}^{3}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Raise $0.91000757$ to the power of $3$ .

$e^{0.91000757} - 1 \cdot 0.75358981$

Step 4.2.2.1.2

Multiply $- 1$ by $0.75358981$ .

$e^{0.91000757} - 0.75358981$

Step 4.2.2.2

Subtract $0.75358981$ from $e^{0.91000757}$ .

$1.73075153$

Step 4.3

Evaluate at $x = 3.73307902$ .

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Step 4.3.1

Substitute $3.73307902$ for $x$ .

$e^{3.73307902} - {(3.73307902)}^{3}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $3.73307902$ to the power of $3$ .

$e^{3.73307902} - 1 \cdot 52.02373776$

Step 4.3.2.1.2

Multiply $- 1$ by $52.02373776$ .

$e^{3.73307902} - 52.02373776$

Step 4.3.2.2

Subtract $52.02373776$ from $e^{3.73307902}$ .

$- 10.21610066$

Step 4.4

List all of the points.

$(- 0.45896226, 0.72861782), (0.91000757, 1.73075153), (3.73307902, - 10.21610066)$

Step 5