Calculus Examples

$y = 3 \sin (\frac{3}{4}) (x + \frac{2 π}{3}) - 5$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Evaluate $\sin (\frac{3}{4})$ .

$\frac{d}{d x} [3 \cdot 0.68163876 (x + \frac{2 π}{3}) - 5]$

Step 1.2

By the Sum Rule, the derivative of $3 \cdot 0.68163876 (x + \frac{2 π}{3}) - 5$ with respect to $x$ is $\frac{d}{d x} [3 \cdot 0.68163876 (x + \frac{2 π}{3})] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [3 \cdot 0.68163876 (x + \frac{2 π}{3})] + \frac{d}{d x} [- 5]$

Step 1.3

Evaluate $\frac{d}{d x} [3 \cdot 0.68163876 (x + \frac{2 π}{3})]$ .

Tap for more steps...

Step 1.3.1

Multiply $3$ by $0.68163876$ .

$\frac{d}{d x} [2.04491628 (x + \frac{2 π}{3})] + \frac{d}{d x} [- 5]$

Step 1.3.2

Since $2.04491628$ is constant with respect to $x$ , the derivative of $2.04491628 (x + \frac{2 π}{3})$ with respect to $x$ is $2.04491628 \frac{d}{d x} [x + \frac{2 π}{3}]$ .

$2.04491628 \frac{d}{d x} [x + \frac{2 π}{3}] + \frac{d}{d x} [- 5]$

Step 1.3.3

By the Sum Rule, the derivative of $x + \frac{2 π}{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{2 π}{3}]$ .

$2.04491628 (\frac{d}{d x} [x] + \frac{d}{d x} [\frac{2 π}{3}]) + \frac{d}{d x} [- 5]$

Step 1.3.4

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

$2.04491628 (1 + \frac{d}{d x} [\frac{2 π}{3}]) + \frac{d}{d x} [- 5]$

Step 1.3.5

Since $\frac{2 π}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 π}{3}$ with respect to $x$ is $0$ .

$2.04491628 (1 + 0) + \frac{d}{d x} [- 5]$

Step 1.3.6

Add $1$ and $0$ .

$2.04491628 \cdot 1 + \frac{d}{d x} [- 5]$

Step 1.3.7

Multiply $2.04491628$ by $1$ .

$2.04491628 + \frac{d}{d x} [- 5]$

Step 1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$2.04491628 + 0$

Step 1.4.2

Add $2.04491628$ and $0$ .

$2.04491628$

Step 2

Since $2.04491628$ is constant with respect to $x$ , the derivative of $2.04491628$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 3

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

$2.04491628 = 0$

Step 4

Since $2.04491628 \neq 0$ , there are no solutions.

No solution

Step 5

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

$0$

Step 6

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

Tap for more steps...

Step 6.1

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

$(- \infty, \infty)$

Step 6.2

Substitute any number, such as $0$ , from the interval $(- \infty, \infty)$ in the first derivative $2.04491628$ to check if the result is negative or positive.

Tap for more steps...

Step 6.2.1

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

$f' (0) = 2.04491628$

Step 6.2.2

The final answer is $2.04491628$ .

$2.04491628$

Step 6.3

No local maxima or minima found for $f (x) = 3 \sin (\frac{3}{4}) (x + \frac{2 π}{3}) - 5$ .

No local maxima or minima

Step 7