Mister Exam
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- 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
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How to use it?
- Derivative of:
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Derivative of y=sin(x)/ln(x)
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Derivative of sinx/(3-2cos(x))
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Derivative of 3y^2
- Derivative of sqrt(r^2-x^2)
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Identical expressions
- y=sin(x)/ln(x)
- y equally sinus of (x) divide by ln(x)
- y=sinx/lnx
- y=sin(x) divide by ln(x)
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Similar expressions
- y=xe^3+sinx/lnx
- (sinx)/(lnx)
- (sinx)/(ln(x+2))
- y=(x^5-2)*sinx/lnxe^x
- y=sinx/ln(x)
Derivative of y=sin(x)/ln(x)
The solution
You have entered
[src]
sin(x) ------ log(x)
$$\frac{\sin{\left(x \right)}}{\log{\left(x \right)}}$$
d /sin(x)\ --|------| dx\log(x)/
$$\frac{d}{d x} \frac{\sin{\left(x \right)}}{\log{\left(x \right)}}$$
Detail solution
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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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The derivative of is .
Now plug in to the quotient rule:
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Now simplify:
The answer is:
The first derivative
[src]
cos(x) sin(x)
------ - ---------
log(x) 2
x*log (x)
$$\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}$$
The second derivative
[src]
/ 2 \
|1 + ------|*sin(x)
2*cos(x) \ log(x)/
-sin(x) - -------- + -------------------
x*log(x) 2
x *log(x)
----------------------------------------
log(x)
$$\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}$$
The third derivative
[src]
/ 3 3 \
2*|1 + ------ + -------|*sin(x) / 2 \
| log(x) 2 | 3*|1 + ------|*cos(x)
3*sin(x) \ log (x)/ \ log(x)/
-cos(x) + -------- - ------------------------------- + ---------------------
x*log(x) 3 2
x *log(x) x *log(x)
----------------------------------------------------------------------------
log(x)
$$\frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{3 \cdot \left(1 + \frac{2}{\log{\left(x \right)}}\right) \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}} - \frac{2 \cdot \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}}}{\log{\left(x \right)}}$$
The graph