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y=sin(15x-sqrtx)
The derivative of the function/ y=sin(15x-sqrtx)

Derivative of y=sin(15x-sqrtx)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   /         ___\
sin\15*x - \/ x /
sin⁡(−x+15x)\sin{\left(- \sqrt{x} + 15 x \right)} 
sin(15*x - sqrt(x))
Detail solution

  1. Let u=−x+15xu = - \sqrt{x} + 15 x.

  2. The derivative of sine is cosine:

    ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddx(−x+15x)\frac{d}{d x} \left(- \sqrt{x} + 15 x\right):

    1. Differentiate −x+15x- \sqrt{x} + 15 x 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: xx goes to 11

        So, the result is: 1515

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

        1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

        So, the result is: −12x- \frac{1}{2 \sqrt{x}}

      The result is: 15−12x15 - \frac{1}{2 \sqrt{x}}

    The result of the chain rule is:

    (15−12x)cos⁡(x−15x)\left(15 - \frac{1}{2 \sqrt{x}}\right) \cos{\left(\sqrt{x} - 15 x \right)}

  4. Now simplify:

    (30x−1)cos⁡(x−15x)2x\frac{\left(30 \sqrt{x} - 1\right) \cos{\left(\sqrt{x} - 15 x \right)}}{2 \sqrt{x}}

The answer is:

(30x−1)cos⁡(x−15x)2x\frac{\left(30 \sqrt{x} - 1\right) \cos{\left(\sqrt{x} - 15 x \right)}}{2 \sqrt{x}}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/        1   \    /  ___       \
|15 - -------|*cos\\/ x  - 15*x/
|         ___|                  
\     2*\/ x /
(15−12x)cos⁡(x−15x)\left(15 - \frac{1}{2 \sqrt{x}}\right) \cos{\left(\sqrt{x} - 15 x \right)}
Simplify
The second derivative [src] 
   /  ___       \               2                  
cos\\/ x  - 15*x/   /       1  \     /  ___       \
----------------- + |30 - -----| *sin\\/ x  - 15*x/
        3/2         |       ___|                   
       x            \     \/ x /                   
---------------------------------------------------
                         4
(30−1x)2sin⁡(x−15x)+cos⁡(x−15x)x324\frac{\left(30 - \frac{1}{\sqrt{x}}\right)^{2} \sin{\left(\sqrt{x} - 15 x \right)} + \frac{\cos{\left(\sqrt{x} - 15 x \right)}}{x^{\frac{3}{2}}}}{4}
Simplify
The third derivative [src] 
                                                            /       1  \    /  ___       \
                                                          3*|30 - -----|*sin\\/ x  - 15*x/
              3                          /  ___       \     |       ___|                  
  /       1  \     /  ___       \   3*cos\\/ x  - 15*x/     \     \/ x /                  
- |30 - -----| *cos\\/ x  - 15*x/ - ------------------- + --------------------------------
  |       ___|                               5/2                         3/2              
  \     \/ x /                              x                           x                 
------------------------------------------------------------------------------------------
                                            8
−(30−1x)3cos⁡(x−15x)+3(30−1x)sin⁡(x−15x)x32−3cos⁡(x−15x)x528\frac{- \left(30 - \frac{1}{\sqrt{x}}\right)^{3} \cos{\left(\sqrt{x} - 15 x \right)} + \frac{3 \left(30 - \frac{1}{\sqrt{x}}\right) \sin{\left(\sqrt{x} - 15 x \right)}}{x^{\frac{3}{2}}} - \frac{3 \cos{\left(\sqrt{x} - 15 x \right)}}{x^{\frac{5}{2}}}}{8}
Simplify
The graph
Derivative of y=sin(15x-sqrtx)
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