Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of e^-1
Derivative of e^x*cosx
Derivative of 6*x^3
Derivative of 5*tan(x)
Identical expressions
y=ln^ two (4x+ five)
y equally ln squared (4x plus 5)
y equally ln to the power of two (4x plus five)
y=ln2(4x+5)
y=ln24x+5
y=ln²(4x+5)
y=ln to the power of 2(4x+5)
y=ln^24x+5
Similar expressions
y=ln^2(4x-5)

The derivative of the function/ y=ln^2(4x+5)

Derivative of y=ln^2(4x+5)

The solution

You have entered [src]

   2         
log (4*x + 5)

$$\log{\left(4 x + 5 \right)}^{2}$$

log(4*x + 5)^2

Detail solution

Let $u = \log{\left(4 x + 5 \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(4 x + 5 \right)}$ :
1. Let $u = 4 x + 5$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 5\right)$ :
  1. Differentiate $4 x + 5$ 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: $4$
    2. The derivative of the constant $5$ is zero.
    The result is: $4$
  The result of the chain rule is:
  
  $\frac{4}{4 x + 5}$
The result of the chain rule is:

$\frac{8 \log{\left(4 x + 5 \right)}}{4 x + 5}$
Now simplify:

$\frac{8 \log{\left(4 x + 5 \right)}}{4 x + 5}$

The answer is:

$\frac{8 \log{\left(4 x + 5 \right)}}{4 x + 5}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

8*log(4*x + 5)
--------------
   4*x + 5

$$\frac{8 \log{\left(4 x + 5 \right)}}{4 x + 5}$$

Simplify

The second derivative [src]

32*(1 - log(5 + 4*x))
---------------------
               2     
      (5 + 4*x)

$$\frac{32 \left(1 - \log{\left(4 x + 5 \right)}\right)}{\left(4 x + 5\right)^{2}}$$

Simplify

The third derivative [src]

128*(-3 + 2*log(5 + 4*x))
-------------------------
                 3       
        (5 + 4*x)

$$\frac{128 \left(2 \log{\left(4 x + 5 \right)} - 3\right)}{\left(4 x + 5\right)^{3}}$$

Simplify

The graph

Derivative of y=ln^2(4x+5)