Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 5-7x
Derivative of 2*sqrt(x^3)
Derivative of ln(x^2-2x+2)
Derivative of (2x-1)/(2x+1)
Identical expressions
log(five *x+ one)^(two)
logarithm of (5 multiply by x plus 1) to the power of (2)
logarithm of (five multiply by x plus one) to the power of (two)
log(5*x+1)(2)
log5*x+12
log(5x+1)^(2)
log(5x+1)(2)
log5x+12
log5x+1^2
Similar expressions
log(5*x-1)^(2)

The derivative of the function/ log(5*x+1)^(2)

Derivative of log(5*x+1)^(2)

The solution

You have entered [src]

   2         
log (5*x + 1)

$$\log{\left(5 x + 1 \right)}^{2}$$

log(5*x + 1)^2

Detail solution

Let $u = \log{\left(5 x + 1 \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(5 x + 1 \right)}$ :
1. Let $u = 5 x + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + 1\right)$ :
  1. Differentiate $5 x + 1$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    2. The derivative of the constant $1$ is zero.
    The result is: $5$
  The result of the chain rule is:
  
  $\frac{5}{5 x + 1}$
The result of the chain rule is:

$\frac{10 \log{\left(5 x + 1 \right)}}{5 x + 1}$
Now simplify:

$\frac{10 \log{\left(5 x + 1 \right)}}{5 x + 1}$

The answer is:

$\frac{10 \log{\left(5 x + 1 \right)}}{5 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

10*log(5*x + 1)
---------------
    5*x + 1

$$\frac{10 \log{\left(5 x + 1 \right)}}{5 x + 1}$$

Simplify

The second derivative [src]

50*(1 - log(1 + 5*x))
---------------------
               2     
      (1 + 5*x)

$$\frac{50 \left(1 - \log{\left(5 x + 1 \right)}\right)}{\left(5 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

250*(-3 + 2*log(1 + 5*x))
-------------------------
                 3       
        (1 + 5*x)

$$\frac{250 \left(2 \log{\left(5 x + 1 \right)} - 3\right)}{\left(5 x + 1\right)^{3}}$$

Simplify

The graph

Derivative of log(5*x+1)^(2)