Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 3*x^5-2*x^2
Derivative of 2^x*sin(x)
Derivative of x*(1-x)
Derivative of sin(3)^(2)*x
Identical expressions
cos(lnx)+5x^ two
co sinus of e of (lnx) plus 5x squared
co sinus of e of (lnx) plus 5x to the power of two
cos(lnx)+5x2
coslnx+5x2
cos(lnx)+5x²
cos(lnx)+5x to the power of 2
coslnx+5x^2
Similar expressions
cos(lnx)-5x^2

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

Derivative of cos(lnx)+5x^2

The solution

You have entered [src]

                 2
cos(log(x)) + 5*x

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

cos(log(x)) + 5*x^2

Detail solution

Differentiate $5 x^{2} + \cos{\left(\log{\left(x \right)} \right)}$ term by term:
1. Let $u = \log{\left(x \right)}$ .
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 $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
4. 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: $10 x$
The result is: $10 x - \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$

The answer is:

$10 x - \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       sin(log(x))
10*x - -----------
            x

$$10 x - \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$$

Simplify

The second derivative [src]

     sin(log(x))   cos(log(x))
10 + ----------- - -----------
           2             2    
          x             x

$$10 + \frac{\sin{\left(\log{\left(x \right)} \right)}}{x^{2}} - \frac{\cos{\left(\log{\left(x \right)} \right)}}{x^{2}}$$

Simplify

The third derivative [src]

-sin(log(x)) + 3*cos(log(x))
----------------------------
              3             
             x

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

Simplify