Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of tan(t)
Derivative of log(x^2+5)
Derivative of e^x*sinx
Derivative of i*n*sin(x)
Identical expressions
а^ two (sin(x)+ one / two (sin2x))
а squared ( sinus of (x) plus 1 divide by 2( sinus of 2x))
а to the power of two ( sinus of (x) plus one divide by two ( sinus of 2x))
а2(sin(x)+1/2(sin2x))
а2sinx+1/2sin2x
а²(sin(x)+1/2(sin2x))
а to the power of 2(sin(x)+1/2(sin2x))
а^2sinx+1/2sin2x
а^2(sin(x)+1 divide by 2(sin2x))
Similar expressions
а^2(sin(x)-1/2(sin2x))
а^2(sinx+1/2(sin2x))

The derivative of the function/ а^2(sin(x)+1/2(sin2x))

Derivative of а^2(sin(x)+1/2(sin2x))

The solution

You have entered [src]

 2 /         sin(2*x)\
a *|sin(x) + --------|
   \            2    /

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

a^2*(sin(x) + sin(2*x)/2)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Differentiate $\sin{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{2}$ term by term:
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 2 x$ .
    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 $\frac{d}{d x} 2 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    So, the result is: $\cos{\left(2 x \right)}$
  The result is: $\cos{\left(x \right)} + \cos{\left(2 x \right)}$
So, the result is: $a^{2} \left(\cos{\left(x \right)} + \cos{\left(2 x \right)}\right)$

The answer is:

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

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  2                      
-a *(4*cos(2*x) + cos(x))

$$- a^{2} \left(\cos{\left(x \right)} + 4 \cos{\left(2 x \right)}\right)$$

Simplify