Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 2*x*sin(x)
Derivative of (x^2+3*x-10)^(5/2)
Derivative of (x+1)*(x+2)^2
Derivative of -x+1
Identical expressions
y= four sin^ four *(x/4)
y equally 4 sinus of to the power of 4 multiply by (x divide by 4)
y equally four sinus of to the power of four multiply by (x divide by 4)
y=4sin4*(x/4)
y=4sin4*x/4
y=4sin⁴*(x/4)
y=4sin^4(x/4)
y=4sin4(x/4)
y=4sin4x/4
y=4sin^4x/4
y=4sin^4*(x divide by 4)

The derivative of the function/ y=4sin^4*(x/4)

Derivative of y=4sin^4*(x/4)

The solution

You have entered [src]

     4/x\
4*sin |-|
      \4/

$$4 \sin^{4}{\left(\frac{x}{4} \right)}$$

d /     4/x\\
--|4*sin |-||
dx\      \4//

$$\frac{d}{d x} 4 \sin^{4}{\left(\frac{x}{4} \right)}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin{\left(\frac{x}{4} \right)}$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{x}{4} \right)}$ :
  1. Let $u = \frac{x}{4}$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{4}$ :
    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: $\frac{1}{4}$
    The result of the chain rule is:
    
    $\frac{\cos{\left(\frac{x}{4} \right)}}{4}$
  The result of the chain rule is:
  
  $\sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}$
So, the result is: $4 \sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}$
Now simplify:

$4 \sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}$

The answer is:

$4 \sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3/x\    /x\
4*sin |-|*cos|-|
      \4/    \4/

$$4 \sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}$$

Simplify

The second derivative [src]

    2/x\ /   2/x\        2/x\\
-sin |-|*|sin |-| - 3*cos |-||
     \4/ \    \4/         \4//

$$- \left(\sin^{2}{\left(\frac{x}{4} \right)} - 3 \cos^{2}{\left(\frac{x}{4} \right)}\right) \sin^{2}{\left(\frac{x}{4} \right)}$$

Simplify

The third derivative [src]

 /       2/x\        2/x\\    /x\    /x\ 
-|- 3*cos |-| + 5*sin |-||*cos|-|*sin|-| 
 \        \4/         \4//    \4/    \4/ 
-----------------------------------------
                    2

$$- \frac{\left(5 \sin^{2}{\left(\frac{x}{4} \right)} - 3 \cos^{2}{\left(\frac{x}{4} \right)}\right) \sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of y=4sin^4*(x/4)

The graph:

Piecewise:

The solution