Mister Exam
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- Derivative of a implicit defined function
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- Partial derivative of the function
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- Differential equations Step by Step
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How to use it?
- Derivative of:
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Derivative of e^x/x^2
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Derivative of 5x^3
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Derivative of 5*x^2-2/sqrt(x)+sin(pi/4)
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Derivative of 2^x+1
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Identical expressions
- ln((sinx)/(cosx+sqrtcosx))
- ln(( sinus of x) divide by ( co sinus of e of x plus square root of co sinus of e of x))
- ln((sinx)/(cosx+√cosx))
- lnsinx/cosx+sqrtcosx
- ln((sinx) divide by (cosx+sqrtcosx))
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Similar expressions
- ln((sinx)/(cosx-sqrtcosx))
Derivative of ln((sinx)/(cosx+sqrtcosx))
The solution
You have entered
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/ sin(x) \ log|-------------------| | ________| \cos(x) + \/ cos(x) /
$$\log{\left(\frac{\sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} \right)}$$
log(sin(x)/(cos(x) + sqrt(cos(x))))
Detail solution
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Let .
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The derivative of is .
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Then, apply the chain rule. Multiply by :
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Apply the quotient rule, which is:
and .
To find :
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The derivative of sine is cosine:
To find :
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Differentiate term by term:
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Let .
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Apply the power rule: goes to
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Then, apply the chain rule. Multiply by :
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The derivative of cosine is negative sine:
The result of the chain rule is:
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The derivative of cosine is negative sine:
The result is:
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Now plug in to the quotient rule:
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The result of the chain rule is:
Now simplify:
The answer is:
The first derivative
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/ / sin(x) \ \
| |------------ + sin(x)|*sin(x)|
| | ________ | |
| cos(x) \2*\/ cos(x) / | / ________\
|------------------- + ------------------------------|*\cos(x) + \/ cos(x) /
| ________ 2 |
|cos(x) + \/ cos(x) / ________\ |
\ \cos(x) + \/ cos(x) / /
----------------------------------------------------------------------------
sin(x)
$$\frac{\left(\frac{\left(\sin{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}}\right) \sin{\left(x \right)}}{\left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}}\right) \left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)}{\sin{\left(x \right)}}$$
The second derivative
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/ 2 / 1 \\ / 2 / 1 \\
| sin (x)*|2 + ----------|| | sin (x)*|2 + ----------||
2 2 | | ________|| | | ________||
________ sin (x) / 1 \ / 1 \ 2 | \ \/ cos(x) /| / 1 \ | \ \/ cos(x) /|
2*\/ cos(x) + 4*cos(x) + --------- |2 + ----------|*cos(x) |2 + ----------| *sin (x) |2*cos(x) + ------------------------|*cos(x) |2 + ----------|*|2*cos(x) + ------------------------|
3/2 | ________| | ________| | ________ | | ________| | ________ |
cos (x) \ \/ cos(x) / \ \/ cos(x) / \ \/ cos(x) + cos(x) / \ \/ cos(x) / \ \/ cos(x) + cos(x) /
-1 + ----------------------------------- + ----------------------- + ------------------------- - -------------------------------------------- - ------------------------------------------------------
/ ________ \ ________ 2 2 / ________ \
4*\\/ cos(x) + cos(x)/ \/ cos(x) + cos(x) / ________ \ 2*sin (x) 4*\\/ cos(x) + cos(x)/
2*\\/ cos(x) + cos(x)/
$$\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right)^{2} \sin^{2}{\left(x \right)}}{2 \left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)^{2}} - \frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \left(\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + 2 \cos{\left(x \right)}\right)}{4 \left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)} + \frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \cos{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} - \frac{\left(\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + 2 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{2 \sin^{2}{\left(x \right)}} - 1 + \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}} + 4 \cos{\left(x \right)}}{4 \left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)}$$
The third derivative
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/ 2 \ 3 / 2 \ / 2 \ 2 / 2 2 \ / 2 2 \
2 | 2 3*sin (x)| 2 / 1 \ / 1 \ 4 | ________ sin (x) | 2 / 1 \ | ________ sin (x) | / 1 \ 2 | ________ sin (x) / 1 \ 2 / 1 \ | | ________ sin (x) / 1 \ 2 / 1 \ |
2 / 1 \ sin (x)*|-8 + ---------- + ---------| 12*sin (x)*|2 + ----------| 6*|2 + ----------| *sin (x) 6*|2*\/ cos(x) + 4*cos(x) + ---------|*cos(x) 6*sin (x)*|2 + ----------|*|2*\/ cos(x) + 4*cos(x) + ---------| 12*|2 + ----------| *sin (x)*cos(x) / 2 / 1 \\ | 2*\/ cos(x) + 4*cos(x) + --------- 2*|2 + ----------| *sin (x) 4*|2 + ----------|*cos(x)| | 2*\/ cos(x) + 4*cos(x) + --------- 2*|2 + ----------| *sin (x) 4*|2 + ----------|*cos(x)| / 2 / 1 \\ / 2 / 1 \\
sin (x)*|2 + ----------| | ________ 5/2 | | ________| | ________| | 3/2 | | ________| | 3/2 | | ________| | sin (x)*|2 + ----------|| | 3/2 | ________| | ________| | | 3/2 | ________| | ________| | | sin (x)*|2 + ----------|| | sin (x)*|2 + ----------||
| ________| \ \/ cos(x) cos (x)/ \ \/ cos(x) / \ \/ cos(x) / \ cos (x)/ \ \/ cos(x) / \ cos (x)/ \ \/ cos(x) / | | ________|| | cos (x) \ \/ cos(x) / \ \/ cos(x) / | / 1 \ | cos (x) \ \/ cos(x) / \ \/ cos(x) / | | | ________|| / 2 \ | | ________||
\ \/ cos(x) / -8*cos(x) + ------------------------------------- - --------------------------- + --------------------------- + ---------------------------------------------- + ---------------------------------------------------------------- + ----------------------------------- 2 | \ \/ cos(x) /| |-4 + ----------------------------------- + --------------------------- + -------------------------|*cos(x) |2 + ----------|*|-4 + ----------------------------------- + --------------------------- + -------------------------|*sin(x) | \ \/ cos(x) /| | ________ sin (x) | / 1 \ | \ \/ cos(x) /|
2*cos(x) + ------------------------ ________ ________ 3 ________ 2 2 cos (x)*|2*cos(x) + ------------------------| | ________ 2 ________ | | ________| | ________ 2 ________ | |2*cos(x) + ------------------------|*|2*\/ cos(x) + 4*cos(x) + ---------| |2 + ----------|*|2*cos(x) + ------------------------|*cos(x)
________ \/ cos(x) + cos(x) \/ cos(x) + cos(x) / ________ \ \/ cos(x) + cos(x) / ________ \ / ________ \ | ________ | | \/ cos(x) + cos(x) / ________ \ \/ cos(x) + cos(x) | \ \/ cos(x) / | \/ cos(x) + cos(x) / ________ \ \/ cos(x) + cos(x) | | ________ | | 3/2 | | ________| | ________ |
\/ cos(x) + cos(x) \\/ cos(x) + cos(x)/ \\/ cos(x) + cos(x)/ \\/ cos(x) + cos(x)/ \ \/ cos(x) + cos(x) / \ \\/ cos(x) + cos(x)/ / \ \\/ cos(x) + cos(x)/ / \ \/ cos(x) + cos(x) / \ cos (x)/ \ \/ cos(x) / \ \/ cos(x) + cos(x) /
----------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------- - ----------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------- + -------------------------------------------------------------
2*sin(x) 8*sin(x) 3 2*sin(x) / ________ \ / ________ \ / ________ \
sin (x) 4*\\/ cos(x) + cos(x)/ 8*\\/ cos(x) + cos(x)/*sin(x) 2*\\/ cos(x) + cos(x)/*sin(x)
$$\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \left(\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + 2 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{2 \left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right) \sin{\left(x \right)}} - \frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \left(\frac{2 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right)^{2} \sin^{2}{\left(x \right)}}{\left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)^{2}} + \frac{4 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \cos{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} - 4 + \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}} + 4 \cos{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}}\right) \sin{\left(x \right)}}{4 \left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)} + \frac{\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + 2 \cos{\left(x \right)}}{2 \sin{\left(x \right)}} + \frac{\left(\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + 2 \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{\left(\frac{\left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + 2 \cos{\left(x \right)}\right) \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}} + 4 \cos{\left(x \right)}\right)}{8 \left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right) \sin{\left(x \right)}} - \frac{\left(\frac{2 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right)^{2} \sin^{2}{\left(x \right)}}{\left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)^{2}} + \frac{4 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \cos{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} - 4 + \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}} + 4 \cos{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}}\right) \cos{\left(x \right)}}{2 \sin{\left(x \right)}} + \frac{\frac{6 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right)^{3} \sin^{4}{\left(x \right)}}{\left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)^{3}} + \frac{12 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right)^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)^{2}} - \frac{12 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + \frac{6 \left(2 + \frac{1}{\sqrt{\cos{\left(x \right)}}}\right) \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}} + 4 \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\left(\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}\right)^{2}} - 8 \cos{\left(x \right)} + \frac{\left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{\frac{5}{2}}{\left(x \right)}} - 8 + \frac{2}{\sqrt{\cos{\left(x \right)}}}\right) \sin^{2}{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}} + \frac{6 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}} + 4 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sqrt{\cos{\left(x \right)}} + \cos{\left(x \right)}}}{8 \sin{\left(x \right)}}$$