Calculus Examples

$f (x) = \cos (x) + 2 x$

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 $\cos (x) + 2 x$ with respect to $x$ is $\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [2 x]$

Step 1.1.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- \sin (x) + \frac{d}{d x} [2 x]$

Step 1.1.3

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

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Step 1.1.3.1

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

$- \sin (x) + 2 \frac{d}{d x} [x]$

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 $n = 1$ .

$- \sin (x) + 2 \cdot 1$

Step 1.1.3.3

Multiply $2$ by $1$ .

$- \sin (x) + 2$

Step 1.1.4

Reorder terms.

$f' (x) = 2 - \sin (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 - \sin (x)$ .

$2 - \sin (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 - \sin (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 - \sin (x) = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

$- \sin (x) = - 2$

Step 2.3

Divide each term in $- \sin (x) = - 2$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- \sin (x) = - 2$ by $- 1$ .

$\frac{- \sin (x)}{- 1} = \frac{- 2}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (x)}{1} = \frac{- 2}{- 1}$

Step 2.3.2.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{- 2}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 2$ by $- 1$ .

$\sin (x) = 2$

Step 2.4

The range of sine is $- 1 \leq y \leq 1$ . Since $2$ does not fall in this range, there is no solution.

No solution

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

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found