Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 3x
Derivative of xln2x
Derivative of ln(cos(x^2))
Derivative of 1/2
Identical expressions
ln(cos(x^ two))
ln( co sinus of e of (x squared ))
ln( co sinus of e of (x to the power of two))
ln(cos(x2))
lncosx2
ln(cos(x²))
ln(cos(x to the power of 2))
lncosx^2
Similar expressions
y=x^3ln(cosx^2)
arcctg(x+2)*ln(cos(x^2+1))

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

Derivative of ln(cos(x^2))

The solution

You have entered [src]

   /   / 2\\
log\cos\x //

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

d /   /   / 2\\\
--\log\cos\x ///
dx

$$\frac{d}{d x} \log{\left(\cos{\left(x^{2} \right)} \right)}$$

Transform

Detail solution

Let $u = \cos{\left(x^{2} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x^{2} \right)}$ :
1. Let $u = x^{2}$ .
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 $\frac{d}{d x} x^{2}$ :
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  The result of the chain rule is:
  
  $- 2 x \sin{\left(x^{2} \right)}$
The result of the chain rule is:

$- \frac{2 x \sin{\left(x^{2} \right)}}{\cos{\left(x^{2} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        / 2\
-2*x*sin\x /
------------
     / 2\   
  cos\x /

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

Simplify

The second derivative [src]

   /          / 2\      2    2/ 2\\
   |   2   sin\x /   2*x *sin \x /|
-2*|2*x  + ------- + -------------|
   |          / 2\         2/ 2\  |
   \       cos\x /      cos \x /  /

$$- 2 \cdot \left(\frac{2 x^{2} \sin^{2}{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}} + 2 x^{2} + \frac{\sin{\left(x^{2} \right)}}{\cos{\left(x^{2} \right)}}\right)$$

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln(cos(x^2))

The graph:

Piecewise:

The solution