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Derivative of:
Derivative of 5*x^2-2/sqrt(x)+sin(pi/4)
Derivative of 2*cos(3*x)
Derivative of 1/3*x^3
Derivative of x*lnx
Identical expressions
eight *pi*sin(pi*x)
8 multiply by Pi multiply by sinus of ( Pi multiply by x)
eight multiply by Pi multiply by sinus of ( Pi multiply by x)
8pisin(pix)
8pisinpix

The derivative of the function/ 8*pi*sin(pi*x)

Derivative of 8pisin(pi*x)

The solution

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8*pi*sin(pi*x)

$$8 \pi \sin{\left(\pi x \right)}$$

(8*pi)*sin(pi*x)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \pi x$ .
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 $\frac{d}{d x} \pi x$ :
  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: $\pi$
  The result of the chain rule is:
  
  $\pi \cos{\left(\pi x \right)}$
So, the result is: $8 \pi^{2} \cos{\left(\pi x \right)}$

The answer is:

$8 \pi^{2} \cos{\left(\pi x \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

    2          
8*pi *cos(pi*x)

$$8 \pi^{2} \cos{\left(\pi x \right)}$$

Simplify

The second derivative [src]

     3          
-8*pi *sin(pi*x)

$$- 8 \pi^{3} \sin{\left(\pi x \right)}$$

Simplify

The third derivative [src]

     4          
-8*pi *cos(pi*x)

$$- 8 \pi^{4} \cos{\left(\pi x \right)}$$

Simplify