Derivative of cos(a*x)

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cos(a*x)

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

cos(a*x)

Detail solution

Let $u = a x$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} a x$ :
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: $a$
The result of the chain rule is:

$- a \sin{\left(a x \right)}$

The answer is:

$- a \sin{\left(a x \right)}$

The first derivative [src]

-a*sin(a*x)

$$- a \sin{\left(a x \right)}$$

Simplify

The second derivative [src]

  2         
-a *cos(a*x)

$$- a^{2} \cos{\left(a x \right)}$$

Simplify

The third derivative [src]

 3         
a *sin(a*x)

$$a^{3} \sin{\left(a x \right)}$$

Simplify