Mister Exam

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Derivative of sin(4x^2)

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   /   2\
sin\4*x /

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

sin(4*x^2)

Detail solution

Let $u = 4 x^{2}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
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:

$8 x \cos{\left(4 x^{2} \right)}$

The answer is:

$8 x \cos{\left(4 x^{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       /   2\
8*x*cos\4*x /

$$8 x \cos{\left(4 x^{2} \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

      /     /   2\      2    /   2\\
-64*x*\3*sin\4*x / + 8*x *cos\4*x //

$$- 64 x \left(8 x^{2} \cos{\left(4 x^{2} \right)} + 3 \sin{\left(4 x^{2} \right)}\right)$$

Simplify

The graph