Mister Exam

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The derivative of the function/ sin(ax)-sin(bx)

Derivative of sin(ax)-sin(bx)

The solution

You have entered [src]

sin(a*x) - sin(b*x)

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

d                      
--(sin(a*x) - sin(b*x))
dx

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

Transform

Detail solution

Differentiate $\sin{\left(a x \right)} - \sin{\left(b x \right)}$ term by term:
1. Let $u = a x$ .
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 $\frac{\partial}{\partial x} a x$ :
  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:
  
  $a \cos{\left(a x \right)}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = b x$ .
  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 $\frac{\partial}{\partial x} b x$ :
    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: $b$
    The result of the chain rule is:
    
    $b \cos{\left(b x \right)}$
  So, the result is: $- b \cos{\left(b x \right)}$
The result is: $a \cos{\left(a x \right)} - b \cos{\left(b x \right)}$

The answer is:

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

The first derivative [src]

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

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

Simplify

The second derivative [src]

 2             2         
b *sin(b*x) - a *sin(a*x)

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

Simplify

The third derivative [src]

 3             3         
b *cos(b*x) - a *cos(a*x)

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

Simplify