Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \sin (2 x)]$

Step 1.2.3

Multiply $2$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (2 x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (2 x)]$ .

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

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.3.3

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

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

Step 1.3.4

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

$2 - (\cos (2 x) (2 \cdot 1))$

Step 1.3.5

Multiply $2$ by $1$ .

$2 - (\cos (2 x) \cdot 2)$

Step 1.3.6

Move $2$ to the left of $\cos (2 x)$ .

$2 - (2 \cdot \cos (2 x))$

Step 1.3.7

Multiply $2$ by $- 1$ .

$2 - 2 \cos (2 x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

$f'' (x) = 0 + \frac{d}{d x} (- 2 \cos (2 x))$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = 0 - 2 \frac{d}{d x} \cos (2 x)$

Step 2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$f'' (x) = 0 - 2 (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (2 x))$

Step 2.2.2.2

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

$f'' (x) = 0 - 2 (- \sin (u) \frac{d}{d x} (2 x))$

Step 2.2.2.3

Replace all occurrences of $u$ with $2 x$ .

$f'' (x) = 0 - 2 (- \sin (2 x) \frac{d}{d x} (2 x))$

Step 2.2.3

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

$f'' (x) = 0 - 2 (- \sin (2 x) (2 \frac{d}{d x} (x)))$

Step 2.2.4

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

$f'' (x) = 0 - 2 (- \sin (2 x) (2 \cdot 1))$

Step 2.2.5

Multiply $2$ by $1$ .

$f'' (x) = 0 - 2 (- \sin (2 x) \cdot 2)$

Step 2.2.6

Multiply $2$ by $- 1$ .

$f'' (x) = 0 - 2 (- 2 \sin (2 x))$

Step 2.2.7

Multiply $- 2$ by $- 2$ .

$f'' (x) = 0 + 4 \sin (2 x)$

Step 2.3

Add $0$ and $4 \sin (2 x)$ .

$f'' (x) = 4 \sin (2 x)$

Step 3

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

$2 - 2 \cos (2 x) = 0$

Step 4

Subtract $2$ from both sides of the equation.

$- 2 \cos (2 x) = - 2$

Step 5

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

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Step 5.1

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

$\frac{- 2 \cos (2 x)}{- 2} = \frac{- 2}{- 2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 2 \cos (2 x)}{- 2} = \frac{- 2}{- 2}$

Step 5.2.1.2

Divide $\cos (2 x)$ by $1$ .

$\cos (2 x) = \frac{- 2}{- 2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 2$ by $- 2$ .

$\cos (2 x) = 1$

Step 6

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$2 x = \arccos (1)$

Step 7

Simplify the right side.

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Step 7.1

The exact value of $\arccos (1)$ is $0$ .

$2 x = 0$

Step 8

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 8.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $0$ by $2$ .

$x = 0$

Step 9

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$2 x = 2 π - 0$

Step 10

Solve for $x$ .

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Step 10.1

Simplify.

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Step 10.1.1

Multiply $- 1$ by $0$ .

$2 x = 2 π + 0$

Step 10.1.2

Add $2 π$ and $0$ .

$2 x = 2 π$

Step 10.2

Divide each term in $2 x = 2 π$ by $2$ and simplify.

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Step 10.2.1

Divide each term in $2 x = 2 π$ by $2$ .

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 10.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{2}$

Step 10.2.3

Simplify the right side.

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Step 10.2.3.1

Cancel the common factor of $2$ .

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Step 10.2.3.1.1

Cancel the common factor.

$x = \frac{2 π}{2}$

Step 10.2.3.1.2

Divide $π$ by $1$ .

$x = π$

Step 11

The solution to the equation $2 x = 0$ .

$x = 0, π$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$4 \sin (2 (0))$

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply $2$ by $0$ .

$4 \sin (0)$

Step 13.2

The exact value of $\sin (0)$ is $0$ .

$4 \cdot 0$

Step 13.3

Multiply $4$ by $0$ .

$0$

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, π) \cup (π, \infty)$

Step 14.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $2 - 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = 2 - 2 \cos (2 (- 2))$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Simplify each term.

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Step 14.2.2.1.1

Multiply $2$ by $- 2$ .

$f' (- 2) = 2 - 2 \cos (- 4)$

Step 14.2.2.1.2

Evaluate $\cos (- 4)$ .

$f' (- 2) = 2 - 2 \cdot - 0.65364362$

Step 14.2.2.1.3

Multiply $- 2$ by $- 0.65364362$ .

$f' (- 2) = 2 + 1.30728724$

Step 14.2.2.2

Add $2$ and $1.30728724$ .

$f' (- 2) = 3.30728724$

Step 14.2.2.3

The final answer is $3.30728724$ .

$3.30728724$

Step 14.3

Substitute any number, such as $2$ , from the interval $(0, π)$ in the first derivative $2 - 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = 2 - 2 \cos (2 (2))$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Simplify each term.

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Step 14.3.2.1.1

Multiply $2$ by $2$ .

$f' (2) = 2 - 2 \cos (4)$

Step 14.3.2.1.2

Evaluate $\cos (4)$ .

$f' (2) = 2 - 2 \cdot - 0.65364362$

Step 14.3.2.1.3

Multiply $- 2$ by $- 0.65364362$ .

$f' (2) = 2 + 1.30728724$

Step 14.3.2.2

Add $2$ and $1.30728724$ .

$f' (2) = 3.30728724$

Step 14.3.2.3

The final answer is $3.30728724$ .

$3.30728724$

Step 14.4

Substitute any number, such as $6$ , from the interval $(π, \infty)$ in the first derivative $2 - 2 \cos (2 x)$ to check if the result is negative or positive.

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Step 14.4.1

Replace the variable $x$ with $6$ in the expression.

$f' (6) = 2 - 2 \cos (2 (6))$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify each term.

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Step 14.4.2.1.1

Multiply $2$ by $6$ .

$f' (6) = 2 - 2 \cos (12)$

Step 14.4.2.1.2

Evaluate $\cos (12)$ .

$f' (6) = 2 - 2 \cdot 0.84385395$

Step 14.4.2.1.3

Multiply $- 2$ by $0.84385395$ .

$f' (6) = 2 - 1.68770791$

Step 14.4.2.2

Subtract $1.68770791$ from $2$ .

$f' (6) = 0.31229208$

Step 14.4.2.3

The final answer is $0.31229208$ .

$0.31229208$

Step 14.5

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.6

No local maxima or minima found for $f (x) = 2 x - \sin (2 x)$ .

No local maxima or minima

Step 15