Mister Exam

Other calculators


y=ln(x+11)^12-12x

Derivative of y=ln(x+11)^12-12x

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   12               
log  (x + 11) - 12*x
12x+log(x+11)12- 12 x + \log{\left(x + 11 \right)}^{12}
log(x + 11)^12 - 12*x
Detail solution
  1. Differentiate 12x+log(x+11)12- 12 x + \log{\left(x + 11 \right)}^{12} term by term:

    1. Let u=log(x+11)u = \log{\left(x + 11 \right)}.

    2. Apply the power rule: u12u^{12} goes to 12u1112 u^{11}

    3. Then, apply the chain rule. Multiply by ddxlog(x+11)\frac{d}{d x} \log{\left(x + 11 \right)}:

      1. Let u=x+11u = x + 11.

      2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

      3. Then, apply the chain rule. Multiply by ddx(x+11)\frac{d}{d x} \left(x + 11\right):

        1. Differentiate x+11x + 11 term by term:

          1. Apply the power rule: xx goes to 11

          2. The derivative of the constant 1111 is zero.

          The result is: 11

        The result of the chain rule is:

        1x+11\frac{1}{x + 11}

      The result of the chain rule is:

      12log(x+11)11x+11\frac{12 \log{\left(x + 11 \right)}^{11}}{x + 11}

    4. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: xx goes to 11

      So, the result is: 12-12

    The result is: 12+12log(x+11)11x+11-12 + \frac{12 \log{\left(x + 11 \right)}^{11}}{x + 11}

  2. Now simplify:

    12(x+log(x+11)1111)x+11\frac{12 \left(- x + \log{\left(x + 11 \right)}^{11} - 11\right)}{x + 11}


The answer is:

12(x+log(x+11)1111)x+11\frac{12 \left(- x + \log{\left(x + 11 \right)}^{11} - 11\right)}{x + 11}

The graph
02468-8-6-4-2-1010-5000001000000
The first derivative [src]
            11        
      12*log  (x + 11)
-12 + ----------------
           x + 11     
12+12log(x+11)11x+11-12 + \frac{12 \log{\left(x + 11 \right)}^{11}}{x + 11}
The second derivative [src]
      10                           
12*log  (11 + x)*(11 - log(11 + x))
-----------------------------------
                     2             
             (11 + x)              
12(11log(x+11))log(x+11)10(x+11)2\frac{12 \left(11 - \log{\left(x + 11 \right)}\right) \log{\left(x + 11 \right)}^{10}}{\left(x + 11\right)^{2}}
The third derivative [src]
      9         /                            2        \
12*log (11 + x)*\110 - 33*log(11 + x) + 2*log (11 + x)/
-------------------------------------------------------
                               3                       
                       (11 + x)                        
12(2log(x+11)233log(x+11)+110)log(x+11)9(x+11)3\frac{12 \left(2 \log{\left(x + 11 \right)}^{2} - 33 \log{\left(x + 11 \right)} + 110\right) \log{\left(x + 11 \right)}^{9}}{\left(x + 11\right)^{3}}
The graph
Derivative of y=ln(x+11)^12-12x