Calculus Examples

$2 \cos (x) + \cos^{2} (x)$

Step 1

Write $2 \cos (x) + \cos^{2} (x)$ as a function.

$f (x) = 2 \cos (x) + \cos^{2} (x)$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

Tap for more steps...

Step 2.1

Find the second derivative.

Tap for more steps...

Step 2.1.1

Find the first derivative.

Tap for more steps...

Step 2.1.1.1

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

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

Step 2.1.1.2

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

Tap for more steps...

Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

Multiply $- 1$ by $2$ .

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

Step 2.1.1.3

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

Tap for more steps...

Step 2.1.1.3.1

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

Tap for more steps...

Step 2.1.1.3.1.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

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

Step 2.1.1.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 2.1.1.3.1.3

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 2.1.1.3.2

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

$- 2 \sin (x) + 2 \cos (x) (- \sin (x))$

Step 2.1.1.3.3

Multiply $- 1$ by $2$ .

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

Step 2.1.1.4

Reorder terms.

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

Step 2.1.2

Find the second derivative.

Tap for more steps...

Step 2.1.2.1

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

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

Step 2.1.2.2

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

Tap for more steps...

Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

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

Step 2.1.2.2.3

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

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

Step 2.1.2.2.4

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

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

Step 2.1.2.2.5

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.2.2.6

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.2.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.2.2.8

Add $1$ and $1$ .

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

Step 2.1.2.2.9

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.2.2.10

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.2.2.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.2.2.12

Add $1$ and $1$ .

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

Step 2.1.2.3

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

Tap for more steps...

Step 2.1.2.3.1

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

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

Step 2.1.2.3.2

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

$- 2 (\cos^{2} (x) - \sin^{2} (x)) - 2 \cos (x)$

Step 2.1.2.4

Simplify.

Tap for more steps...

Step 2.1.2.4.1

Apply the distributive property.

$- 2 \cos^{2} (x) - 2 (- \sin^{2} (x)) - 2 \cos (x)$

Step 2.1.2.4.2

Multiply $- 1$ by $- 2$ .

$f'' (x) = - 2 \cos^{2} (x) + 2 \sin^{2} (x) - 2 \cos (x)$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 2 \cos^{2} (x) + 2 \sin^{2} (x) - 2 \cos (x)$ .

$- 2 \cos^{2} (x) + 2 \sin^{2} (x) - 2 \cos (x)$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- 2 \cos^{2} (x) + 2 \sin^{2} (x) - 2 \cos (x) = 0$ .

Tap for more steps...

Step 2.2.1

Set the second derivative equal to $0$ .

$- 2 \cos^{2} (x) + 2 \sin^{2} (x) - 2 \cos (x) = 0$

Step 2.2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{π}{3} + \frac{2 π n}{3}$ , for any integer $n$

Step 3

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, \infty)$

Step 5

Substitute any number from the interval $(- \infty, \infty)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = - 2 \cos^{2} (0) + 2 \sin^{2} (0) - 2 \cos (0)$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

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

$f'' (0) = - 2 \cdot 1^{2} + 2 \sin^{2} (0) - 2 \cos (0)$

Step 5.2.1.2

One to any power is one.

$f'' (0) = - 2 \cdot 1 + 2 \sin^{2} (0) - 2 \cos (0)$

Step 5.2.1.3

Multiply $- 2$ by $1$ .

$f'' (0) = - 2 + 2 \sin^{2} (0) - 2 \cos (0)$

Step 5.2.1.4

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

$f'' (0) = - 2 + 2 \cdot 0^{2} - 2 \cos (0)$

Step 5.2.1.5

Raising $0$ to any positive power yields $0$ .

$f'' (0) = - 2 + 2 \cdot 0 - 2 \cos (0)$

Step 5.2.1.6

Multiply $2$ by $0$ .

$f'' (0) = - 2 + 0 - 2 \cos (0)$

Step 5.2.1.7

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

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

Step 5.2.1.8

Multiply $- 2$ by $1$ .

$f'' (0) = - 2 + 0 - 2$

Step 5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 5.2.2.1

Add $- 2$ and $0$ .

$f'' (0) = - 2 - 2$

Step 5.2.2.2

Subtract $2$ from $- 2$ .

$f'' (0) = - 4$

Step 5.2.3

The final answer is $- 4$ .

$- 4$

Step 5.3

The graph is concave down on the interval $(- \infty, \infty)$ because $f'' (0)$ is negative.

Concave down on $(- \infty, \infty)$ since $f'' (x)$ is negative

Step 6