Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of tan(x)*sin(x)
Derivative of y=tgx-cosx
Derivative of ctg(5x)
Derivative of sec^2(x^2)
Identical expressions
sec^ two (x^ two)
sec squared (x squared )
sec to the power of two (x to the power of two)
sec2(x2)
sec2x2
sec²(x²)
sec to the power of 2(x to the power of 2)
sec^2x^2

The derivative of the function/ sec^2(x^2)

Derivative of sec^2(x^2)

The solution

You have entered [src]

   2/ 2\
sec \x /

$$\sec^{2}{\left(x^{2} \right)}$$

sec(x^2)^2

Detail solution

Let $u = \sec{\left(x^{2} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sec{\left(x^{2} \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\sec{\left(x^{2} \right)} = \frac{1}{\cos{\left(x^{2} \right)}}$
2. Let $u = \cos{\left(x^{2} \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. 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:
  
  $\frac{2 x \sin{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}$
The result of the chain rule is:

$\frac{4 x \sin{\left(x^{2} \right)} \sec{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2/ 2\    / 2\
4*x*sec \x /*tan\x /

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

Simplify

The second derivative [src]

     2/ 2\ /   2 /       2/ 2\\      2    2/ 2\      / 2\\
4*sec \x /*\2*x *\1 + tan \x // + 4*x *tan \x / + tan\x //

$$4 \left(2 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} \right)} + \tan{\left(x^{2} \right)}\right) \sec^{2}{\left(x^{2} \right)}$$

Simplify

The third derivative [src]

       2/ 2\ /         2/ 2\      2    3/ 2\       2 /       2/ 2\\    / 2\\
8*x*sec \x /*\3 + 9*tan \x / + 8*x *tan \x / + 16*x *\1 + tan \x //*tan\x //

$$8 x \left(16 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \tan{\left(x^{2} \right)} + 8 x^{2} \tan^{3}{\left(x^{2} \right)} + 9 \tan^{2}{\left(x^{2} \right)} + 3\right) \sec^{2}{\left(x^{2} \right)}$$

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of sec^2(x^2)

The graph:

Piecewise:

The solution