Derivative of y=sin(15x-sqrtx)

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   /         ___\
sin\15*x - \/ x /

$$\sin{\left(- \sqrt{x} + 15 x \right)}$$

sin(15*x - sqrt(x))

Detail solution

Let $u = - \sqrt{x} + 15 x$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- \sqrt{x} + 15 x\right)$ :
1. Differentiate $- \sqrt{x} + 15 x$ 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: $x$ goes to $1$
    So, the result is: $15$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    So, the result is: $- \frac{1}{2 \sqrt{x}}$
  The result is: $15 - \frac{1}{2 \sqrt{x}}$
The result of the chain rule is:

$\left(15 - \frac{1}{2 \sqrt{x}}\right) \cos{\left(\sqrt{x} - 15 x \right)}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

/        1   \    /  ___       \
|15 - -------|*cos\\/ x  - 15*x/
|         ___|                  
\     2*\/ x /

$$\left(15 - \frac{1}{2 \sqrt{x}}\right) \cos{\left(\sqrt{x} - 15 x \right)}$$

Simplify

The second derivative [src]

   /  ___       \               2                  
cos\\/ x  - 15*x/   /       1  \     /  ___       \
----------------- + |30 - -----| *sin\\/ x  - 15*x/
        3/2         |       ___|                   
       x            \     \/ x /                   
---------------------------------------------------
                         4

$$\frac{\left(30 - \frac{1}{\sqrt{x}}\right)^{2} \sin{\left(\sqrt{x} - 15 x \right)} + \frac{\cos{\left(\sqrt{x} - 15 x \right)}}{x^{\frac{3}{2}}}}{4}$$

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

$$\frac{- \left(30 - \frac{1}{\sqrt{x}}\right)^{3} \cos{\left(\sqrt{x} - 15 x \right)} + \frac{3 \left(30 - \frac{1}{\sqrt{x}}\right) \sin{\left(\sqrt{x} - 15 x \right)}}{x^{\frac{3}{2}}} - \frac{3 \cos{\left(\sqrt{x} - 15 x \right)}}{x^{\frac{5}{2}}}}{8}$$

Simplify

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Derivative of y=sin(15x-sqrtx)

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