Calculus Examples

$f (x) = (9 x + 5) e^{- 6 x}$

Step 1

Set $(9 x + 5) e^{- 6 x}$ equal to $0$ .

$(9 x + 5) e^{- 6 x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$9 x + 5 = 0$

$e^{- 6 x} = 0$

Step 2.2

Set $9 x + 5$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $9 x + 5$ equal to $0$ .

$9 x + 5 = 0$

Step 2.2.2

Solve $9 x + 5 = 0$ for $x$ .

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Step 2.2.2.1

Subtract $5$ from both sides of the equation.

$9 x = - 5$

Step 2.2.2.2

Divide each term in $9 x = - 5$ by $9$ and simplify.

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Step 2.2.2.2.1

Divide each term in $9 x = - 5$ by $9$ .

$\frac{9 x}{9} = \frac{- 5}{9}$

Step 2.2.2.2.2

Simplify the left side.

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Step 2.2.2.2.2.1

Cancel the common factor of $9$ .

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Step 2.2.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{- 5}{9}$

Step 2.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{9}$

Step 2.2.2.2.3

Simplify the right side.

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Step 2.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{9}$

Step 2.3

Set $e^{- 6 x}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $e^{- 6 x}$ equal to $0$ .

$e^{- 6 x} = 0$

Step 2.3.2

Solve $e^{- 6 x} = 0$ for $x$ .

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Step 2.3.2.1

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

$\ln (e^{- 6 x}) = \ln (0)$

Step 2.3.2.2

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

Undefined

Step 2.3.2.3

There is no solution for $e^{- 6 x} = 0$

No solution

Step 2.4

The final solution is all the values that make $(9 x + 5) e^{- 6 x} = 0$ true.

$x = - \frac{5}{9}$

Step 3