Calculus Examples

$y = - 8 x + e^{x}$

Step 1

Set $y$ as a function of $x$ .

$f (x) = - 8 x + e^{x}$

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 8 x + e^{x}$ with respect to $x$ is $\frac{d}{d x} [- 8 x] + \frac{d}{d x} [e^{x}]$ .

$\frac{d}{d x} [- 8 x] + \frac{d}{d x} [e^{x}]$

Step 2.2

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 2.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$- 8 \frac{d}{d x} [x] + \frac{d}{d x} [e^{x}]$

Step 2.2.2

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

$- 8 \cdot 1 + \frac{d}{d x} [e^{x}]$

Step 2.2.3

Multiply $- 8$ by $1$ .

$- 8 + \frac{d}{d x} [e^{x}]$

Step 2.3

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

$- 8 + e^{x}$

Step 3

Set the derivative equal to $0$ then solve the equation $- 8 + e^{x} = 0$ .

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Step 3.1

Add $8$ to both sides of the equation.

$e^{x} = 8$

Step 3.2

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

$\ln (e^{x}) = \ln (8)$

Step 3.3

Expand the left side.

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Step 3.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (8)$

Step 3.3.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (8)$

Step 3.3.3

Multiply $x$ by $1$ .

$x = \ln (8)$

Step 4

Solve the original function $f (x) = - 8 x + e^{x}$ at $x = \ln (8)$ .

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Step 4.1

Replace the variable $x$ with $\ln (8)$ in the expression.

$f (\ln (8)) = - 8 \ln (8) + e^{\ln (8)}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Simplify $- 8 \ln (8)$ by moving $8$ inside the logarithm.

$f (\ln (8)) = - \ln (8^{8}) + e^{\ln (8)}$

Step 4.2.1.2

Raise $8$ to the power of $8$ .

$f (\ln (8)) = - \ln (16777216) + e^{\ln (8)}$

Step 4.2.1.3

Exponentiation and log are inverse functions.

$f (\ln (8)) = - \ln (16777216) + 8$

Step 4.2.2

The final answer is $- \ln (16777216) + 8$ .

$- \ln (16777216) + 8$

Step 5

The horizontal tangent line on function $f (x) = - 8 x + e^{x}$ is $y = - \ln (16777216) + 8$ .

$y = - \ln (16777216) + 8$

Step 6