Calculus Examples

$\frac{d}{d x} e^{x^{3} + 2} = e^{x^{3} + 2}$

Step 1

Simplify $\frac{d}{d x} \cdot e^{x^{3} + 2}$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{d}{d x} \cdot e^{x^{3} + 2} = e^{x^{3} + 2}$

Step 1.2

Simplify by adding zeros.

$\frac{d}{d x} \cdot e^{x^{3} + 2} = e^{x^{3} + 2}$

Step 1.3

Cancel the common factor of $d$ .

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Step 1.3.1

Cancel the common factor.

$\frac{d}{d x} \cdot e^{x^{3} + 2} = e^{x^{3} + 2}$

Step 1.3.2

Rewrite the expression.

$\frac{1}{x} \cdot e^{x^{3} + 2} = e^{x^{3} + 2}$

Step 1.4

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

$\frac{e^{x^{3} + 2}}{x} = e^{x^{3} + 2}$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $e^{x^{3} + 2}$ from both sides of the equation.

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

Step 2.2

To write $- e^{x^{3} + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2.3

Combine $- e^{x^{3} + 2}$ and $\frac{x}{x}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Factor $e^{x^{3} + 2}$ out of $e^{x^{3} + 2} - e^{x^{3} + 2} x$ .

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Step 2.5.1

Multiply by $1$ .

$\frac{e^{x^{3} + 2} \cdot 1 - e^{x^{3} + 2} x}{x} = 0$

Step 2.5.2

Factor $e^{x^{3} + 2}$ out of $- e^{x^{3} + 2} x$ .

$\frac{e^{x^{3} + 2} \cdot 1 + e^{x^{3} + 2} (- x)}{x} = 0$

Step 2.5.3

Factor $e^{x^{3} + 2}$ out of $e^{x^{3} + 2} \cdot 1 + e^{x^{3} + 2} (- x)$ .

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

Step 3

Set the numerator equal to zero.

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

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

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

$1 - x = 0$

Step 4.2

Set $e^{x^{3} + 2}$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $e^{x^{3} + 2}$ equal to $0$ .

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

Step 4.2.2

Solve $e^{x^{3} + 2} = 0$ for $x$ .

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Step 4.2.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x^{3} + 2}) = \ln (0)$

Step 4.2.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 4.2.2.3

There is no solution for $e^{x^{3} + 2} = 0$

No solution

Step 4.3

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 4.3.2

Solve $1 - x = 0$ for $x$ .

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Step 4.3.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 4.3.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 4.3.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 4.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 4.4

The final solution is all the values that make $e^{x^{3} + 2} (1 - x) = 0$ true.

$x = 1$

Step 5

To rewrite as a function of $d$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $d$ is on the other side.

$f (d) = 1$