Mister Exam
Other calculators
- Derivative of a implicit defined function
- Derivative of Parametric Function
- Partial derivative of the function
- Curve tracing functions Step by Step
- Integral Step by Step
- Differential equations Step by Step
- Limits Step by Step
-
How to use it?
- Derivative of:
-
Derivative of e^-1
-
Derivative of 2*x^5
-
Derivative of sin(2*x^2+3)
-
Derivative of f(x)=5
-
Identical expressions
- √ln(x+√x+√x²+sin2x)
- √ln(x plus √x plus √x² plus sinus of 2x)
- √lnx+√x+√x²+sin2x
-
Similar expressions
- √ln(x-√x+√x²+sin2x)
- √ln(x+√x+√x²-sin2x)
- √ln(x+√x-√x²+sin2x)
Derivative of √ln(x+√x+√x²+sin2x)
The solution
You have entered
[src]
____________________________________ / / 2 \ / | ___ ___ | \/ log\x + \/ x + \/ x + sin(2*x)/
$$\sqrt{\log{\left(\left(\left(\sqrt{x}\right)^{2} + \left(\sqrt{x} + x\right)\right) + \sin{\left(2 x \right)} \right)}}$$
sqrt(log(x + sqrt(x) + (sqrt(x))^2 + sin(2*x)))
Detail solution
-
Let .
-
Apply the power rule: goes to
-
Then, apply the chain rule. Multiply by :
-
Let .
-
The derivative of is .
-
-
Then, apply the chain rule. Multiply by :
-
Differentiate term by term:
-
Differentiate term by term:
-
Differentiate term by term:
-
Apply the power rule: goes to
-
Apply the power rule: goes to
The result is:
-
-
Let .
-
Apply the power rule: goes to
-
-
Then, apply the chain rule. Multiply by :
-
Apply the power rule: goes to
The result of the chain rule is:
-
The result is:
-
-
Let .
-
The derivative of sine is cosine:
-
-
Then, apply the chain rule. Multiply by :
-
The derivative of a constant times a function is the constant times the derivative of the function.
-
Apply the power rule: goes to
So, the result is:
-
The result of the chain rule is:
-
The result is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
The first derivative
[src]
1 x
1 + ------- + 2*cos(2*x) + -
___ x
2*\/ x
--------------------------------------------------------------------------
____________________________________
/ 2 \ / / 2 \
| ___ ___ | / | ___ ___ |
2*\x + \/ x + \/ x + sin(2*x)/*\/ log\x + \/ x + \/ x + sin(2*x)/
$$\frac{2 \cos{\left(2 x \right)} + 1 + \frac{x}{x} + \frac{1}{2 \sqrt{x}}}{2 \left(\left(\left(\sqrt{x}\right)^{2} + \left(\sqrt{x} + x\right)\right) + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\left(\left(\sqrt{x}\right)^{2} + \left(\sqrt{x} + x\right)\right) + \sin{\left(2 x \right)} \right)}}}$$
The second derivative
[src]
/ 2 2 \
| / 1 \ / 1 \ |
| |4 + ----- + 4*cos(2*x)| |4 + ----- + 4*cos(2*x)| |
| | ___ | | ___ | |
| 1 \ \/ x / \ \/ x / |
-|------ + 8*sin(2*x) + -------------------------- + ------------------------------------------------------|
| 3/2 / ___ \ / ___ \ / ___ \|
\2*x 2*\\/ x + 2*x + sin(2*x)/ 4*\\/ x + 2*x + sin(2*x)/*log\\/ x + 2*x + sin(2*x)//
-------------------------------------------------------------------------------------------------------------
_____________________________
/ ___ \ / / ___ \
4*\\/ x + 2*x + sin(2*x)/*\/ log\\/ x + 2*x + sin(2*x)/
$$- \frac{8 \sin{\left(2 x \right)} + \frac{\left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{2}}{2 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)} + \frac{\left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{2}}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}} + \frac{1}{2 x^{\frac{3}{2}}}}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}}}$$
The third derivative
[src]
3 3 3
/ 1 \ / 1 \ / 1 \ / 1 \ / 1 \ / 1 \ / 1 \
|4 + ----- + 4*cos(2*x)| 3*|---- + 16*sin(2*x)|*|4 + ----- + 4*cos(2*x)| 3*|4 + ----- + 4*cos(2*x)| 3*|4 + ----- + 4*cos(2*x)| 3*|---- + 16*sin(2*x)|*|4 + ----- + 4*cos(2*x)|
| ___ | | 3/2 | | ___ | | ___ | | ___ | | 3/2 | | ___ |
3 \ \/ x / \x / \ \/ x / \ \/ x / \ \/ x / \x / \ \/ x /
-32*cos(2*x) + ------ + ------------------------- + ----------------------------------------------- + ------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------
5/2 2 / ___ \ 2 2 / ___ \ / ___ \
2*x / ___ \ 2*\\/ x + 2*x + sin(2*x)/ / ___ \ / ___ \ / ___ \ 2/ ___ \ 4*\\/ x + 2*x + sin(2*x)/*log\\/ x + 2*x + sin(2*x)/
\\/ x + 2*x + sin(2*x)/ 4*\\/ x + 2*x + sin(2*x)/ *log\\/ x + 2*x + sin(2*x)/ 8*\\/ x + 2*x + sin(2*x)/ *log \\/ x + 2*x + sin(2*x)/
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
_____________________________
/ ___ \ / / ___ \
8*\\/ x + 2*x + sin(2*x)/*\/ log\\/ x + 2*x + sin(2*x)/
$$\frac{\frac{3 \left(16 \sin{\left(2 x \right)} + \frac{1}{x^{\frac{3}{2}}}\right) \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)}{2 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)} + \frac{3 \left(16 \sin{\left(2 x \right)} + \frac{1}{x^{\frac{3}{2}}}\right) \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}} - 32 \cos{\left(2 x \right)} + \frac{\left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{3}}{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)^{2}} + \frac{3 \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{3}}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)^{2} \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}} + \frac{3 \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{3}}{8 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)^{2} \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}^{2}} + \frac{3}{2 x^{\frac{5}{2}}}}{8 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}}}$$