Derivative of cos(ax+b)

You have entered [src]

cos(a*x + b)

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

d               
--(cos(a*x + b))
dx

$$\frac{\partial}{\partial x} \cos{\left(a x + b \right)}$$

Transform

Detail solution

Let $u = a x + b$ .
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 $\frac{\partial}{\partial x} \left(a x + b\right)$ :
1. Differentiate $a x + b$ 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: $a$
  2. The derivative of the constant $b$ is zero.
  The result is: $a$
The result of the chain rule is:

$- a \sin{\left(a x + b \right)}$
Now simplify:

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

The answer is:

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

The first derivative [src]

-a*sin(a*x + b)

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

Simplify

The second derivative [src]

  2             
-a *cos(b + a*x)

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

Simplify

The third derivative [src]

 3             
a *sin(b + a*x)

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

Simplify