Calculus Examples

$e^{x} - 2 x$

Step 1

Write $e^{x} - 2 x$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $e^{x} - 2 x$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 2 x]$

Step 2.2

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

$e^{x} + \frac{d}{d x} [- 2 x]$

Step 2.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$e^{x} - 2 \frac{d}{d x} [x]$

Step 2.3.2

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

$e^{x} - 2 \cdot 1$

Step 2.3.3

Multiply $- 2$ by $1$ .

$e^{x} - 2$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $e^{x} - 2$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 2]$ .

$f'' (x) = \frac{d}{d x} (e^{x}) + \frac{d}{d x} (- 2)$

Step 3.2

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

$f'' (x) = e^{x} + \frac{d}{d x} (- 2)$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = e^{x} + 0$

Step 3.3.2

Add $e^{x}$ and $0$ .

$f'' (x) = e^{x}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$e^{x} - 2 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $e^{x} - 2 x$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 2 x]$

Step 5.1.2

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

$e^{x} + \frac{d}{d x} [- 2 x]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 5.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$e^{x} - 2 \frac{d}{d x} [x]$

Step 5.1.3.2

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

$e^{x} - 2 \cdot 1$

Step 5.1.3.3

Multiply $- 2$ by $1$ .

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

Step 5.2

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

$e^{x} - 2$

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

$e^{x} - 2 = 0$

Step 6.2

Add $2$ to both sides of the equation.

$e^{x} = 2$

Step 6.3

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

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

Step 6.4

Expand the left side.

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Step 6.4.1

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

$x \ln (e) = \ln (2)$

Step 6.4.2

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

$x \cdot 1 = \ln (2)$

Step 6.4.3

Multiply $x$ by $1$ .

$x = \ln (2)$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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 8

Critical points to evaluate.

$x = \ln (2)$

Step 9

Evaluate the second derivative at $x = \ln (2)$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$e^{\ln (2)}$

Step 10

Exponentiation and log are inverse functions.

$2$

Step 11

$x = \ln (2)$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \ln (2)$ is a local minimum

Step 12

Find the y-value when $x = \ln (2)$ .

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Step 12.1

Replace the variable $x$ with $\ln (2)$ in the expression.

$f (\ln (2)) = e^{\ln (2)} - 2 \ln (2)$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Exponentiation and log are inverse functions.

$f (\ln (2)) = 2 - 2 \ln (2)$

Step 12.2.1.2

Simplify $- 2 \ln (2)$ by moving $2$ inside the logarithm.

$f (\ln (2)) = 2 - \ln (2^{2})$

Step 12.2.1.3

Raise $2$ to the power of $2$ .

$f (\ln (2)) = 2 - \ln (4)$

Step 12.2.2

The final answer is $2 - \ln (4)$ .

$y = 2 - \ln (4)$

Step 13

These are the local extrema for $f (x) = e^{x} - 2 x$ .

$(\ln (2), 2 - \ln (4))$ is a local minima

Step 14