Mister Exam

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sin(3*x+pi/4)

sin((3*x + pi)/4)

sin(3*(x + pi)/4)

sin(3*x + pi/4)

sin(3*x + pi/4)

Derivative of sin(3*x+pi/4)

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   /      pi\
sin|3*x + --|
   \      4 /

$$\sin{\left(3 x + \frac{\pi}{4} \right)}$$

Transform

Detail solution

Let $u = 3 x + \frac{\pi}{4}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + \frac{\pi}{4}\right)$ :
1. Differentiate $3 x + \frac{\pi}{4}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  2. The derivative of the constant $\frac{\pi}{4}$ is zero.
  The result is: $3$
The result of the chain rule is:

$3 \cos{\left(3 x + \frac{\pi}{4} \right)}$
Now simplify:

$3 \cos{\left(3 x + \frac{\pi}{4} \right)}$

The answer is:

$3 \cos{\left(3 x + \frac{\pi}{4} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     /      pi\
3*cos|3*x + --|
     \      4 /

$$3 \cos{\left(3 x + \frac{\pi}{4} \right)}$$

Simplify

The second derivative [src]

      /      pi\
-9*sin|3*x + --|
      \      4 /

$$- 9 \sin{\left(3 x + \frac{\pi}{4} \right)}$$

Simplify

The third derivative [src]

       /      pi\
-27*cos|3*x + --|
       \      4 /

$$- 27 \cos{\left(3 x + \frac{\pi}{4} \right)}$$

Simplify

The graph