Derivative of sin(pi*x)

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sin(pi*x)

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

sin(pi*x)

Detail solution

Let $u = \pi x$ .
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 $\frac{d}{d x} \pi x$ :
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:

$\pi \cos{\left(\pi x \right)}$

The answer is:

$\pi \cos{\left(\pi x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

pi*cos(pi*x)

$$\pi \cos{\left(\pi x \right)}$$

Simplify

The second derivative [src]

   2          
-pi *sin(pi*x)

$$- \pi^{2} \sin{\left(\pi x \right)}$$

Simplify

The third derivative [src]

   3          
-pi *cos(pi*x)

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

Simplify

The graph