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Derivative of:
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Identical expressions
3cos(two x^2)+ one
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3 co sinus of e of (two x squared ) plus one
3cos(2x2)+1
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Similar expressions
3cos(2x^2)-1

The derivative of the function/ 3cos(2x^2)+1

Derivative of 3cos(2x^2)+1

The solution

You have entered [src]

     /   2\    
3*cos\2*x / + 1

$$3 \cos{\left(2 x^{2} \right)} + 1$$

3*cos(2*x^2) + 1

Detail solution

Differentiate $3 \cos{\left(2 x^{2} \right)} + 1$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 2 x^{2}$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x^{2}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    The result of the chain rule is:
    
    $- 4 x \sin{\left(2 x^{2} \right)}$
  So, the result is: $- 12 x \sin{\left(2 x^{2} \right)}$
2. The derivative of the constant $1$ is zero.
The result is: $- 12 x \sin{\left(2 x^{2} \right)}$

The answer is:

$- 12 x \sin{\left(2 x^{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         /   2\
-12*x*sin\2*x /

$$- 12 x \sin{\left(2 x^{2} \right)}$$

Simplify

The second derivative [src]

    /   2    /   2\      /   2\\
-12*\4*x *cos\2*x / + sin\2*x //

$$- 12 \left(4 x^{2} \cos{\left(2 x^{2} \right)} + \sin{\left(2 x^{2} \right)}\right)$$

Simplify

The third derivative [src]

     /       /   2\      2    /   2\\
48*x*\- 3*cos\2*x / + 4*x *sin\2*x //

$$48 x \left(4 x^{2} \sin{\left(2 x^{2} \right)} - 3 \cos{\left(2 x^{2} \right)}\right)$$

Simplify