Calculus Examples

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

Step 1

Find the first derivative of the function.

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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

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

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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.

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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.

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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.

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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