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Derivative of:
Derivative of e^x/x^2
Derivative of cos(x)/x^2
Derivative of cos(sqrt(x))
Derivative of cos(1/x)
Identical expressions
5cos(4x^ two)
5 co sinus of e of (4x squared )
5 co sinus of e of (4x to the power of two)
5cos(4x2)
5cos4x2
5cos(4x²)
5cos(4x to the power of 2)
5cos4x^2

The derivative of the function/ 5cos(4x^2)

Derivative of 5cos(4x^2)

The solution

You have entered [src]

     /   2\
5*cos\4*x /

$$5 \cos{\left(4 x^{2} \right)}$$

5*cos(4*x^2)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = 4 x^{2}$ .
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 $\frac{d}{d x} 4 x^{2}$ :
  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: $8 x$
  The result of the chain rule is:
  
  $- 8 x \sin{\left(4 x^{2} \right)}$
So, the result is: $- 40 x \sin{\left(4 x^{2} \right)}$

The answer is:

$- 40 x \sin{\left(4 x^{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         /   2\
-40*x*sin\4*x /

$$- 40 x \sin{\left(4 x^{2} \right)}$$

Simplify

The second derivative [src]

    /   2    /   2\      /   2\\
-40*\8*x *cos\4*x / + sin\4*x //

$$- 40 \left(8 x^{2} \cos{\left(4 x^{2} \right)} + \sin{\left(4 x^{2} \right)}\right)$$

Simplify

The third derivative [src]

      /       /   2\      2    /   2\\
320*x*\- 3*cos\4*x / + 8*x *sin\4*x //

$$320 x \left(8 x^{2} \sin{\left(4 x^{2} \right)} - 3 \cos{\left(4 x^{2} \right)}\right)$$

Simplify