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
- Limits Step by Step
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How to use it?
- Derivative of:
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Derivative of x*2^x
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Derivative of (x^2)/36
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Derivative of x^2/log(x)
- Derivative of e^(sin(x)-2*x^3)
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Identical expressions
- ln(sin(x- one / four *cos(x)))
- ln( sinus of (x minus 1 divide by 4 multiply by co sinus of e of (x)))
- ln( sinus of (x minus one divide by four multiply by co sinus of e of (x)))
- ln(sin(x-1/4cos(x)))
- lnsinx-1/4cosx
- ln(sin(x-1 divide by 4*cos(x)))
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Similar expressions
- ln((sin(x)-1/4cos(x)))
- ln(sin(x+1/4*cos(x)))
- ln(sin(x-1/4*cosx))
Derivative of ln(sin(x-1/4*cos(x)))
The solution
You have entered
[src]
/ / cos(x)\\ log|sin|x - ------|| \ \ 4 //
$$\log{\left(\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)} \right)}$$
log(sin(x - cos(x)/4))
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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Let .
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The derivative of sine is cosine:
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Then, apply the chain rule. Multiply by :
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Differentiate term by term:
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Apply the power rule: goes to
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The derivative of a constant times a function is the constant times the derivative of the function.
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The derivative of cosine is negative sine:
So, the result is:
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The result is:
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The result of the chain rule is:
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The result of the chain rule is:
Now simplify:
The answer is:
The first derivative
[src]
/ sin(x)\ / cos(x)\
|1 + ------|*cos|x - ------|
\ 4 / \ 4 /
----------------------------
/ cos(x)\
sin|x - ------|
\ 4 /
$$\frac{\left(\frac{\sin{\left(x \right)}}{4} + 1\right) \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}$$
The second derivative
[src]
2 2/ cos(x)\ / cos(x)\
(4 + sin(x)) *cos |x - ------| 4*cos(x)*cos|x - ------|
2 \ 4 / \ 4 /
- (4 + sin(x)) - ------------------------------ + ------------------------
2/ cos(x)\ / cos(x)\
sin |x - ------| sin|x - ------|
\ 4 / \ 4 /
---------------------------------------------------------------------------
16
$$\frac{- \left(\sin{\left(x \right)} + 4\right)^{2} - \frac{\left(\sin{\left(x \right)} + 4\right)^{2} \cos^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}}{16}$$
The third derivative
[src]
3 3/ cos(x)\ 3 / cos(x)\ / cos(x)\ 2/ cos(x)\
(4 + sin(x)) *cos |x - ------| (4 + sin(x)) *cos|x - ------| 8*cos|x - ------|*sin(x) 6*cos |x - ------|*(4 + sin(x))*cos(x)
\ 4 / \ 4 / \ 4 / \ 4 /
-6*(4 + sin(x))*cos(x) + ------------------------------ + ----------------------------- - ------------------------ - --------------------------------------
3/ cos(x)\ / cos(x)\ / cos(x)\ 2/ cos(x)\
sin |x - ------| sin|x - ------| sin|x - ------| sin |x - ------|
\ 4 / \ 4 / \ 4 / \ 4 /
-----------------------------------------------------------------------------------------------------------------------------------------------------------
32
$$\frac{\frac{\left(\sin{\left(x \right)} + 4\right)^{3} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} + \frac{\left(\sin{\left(x \right)} + 4\right)^{3} \cos^{3}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{3}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} - 6 \left(\sin{\left(x \right)} + 4\right) \cos{\left(x \right)} - \frac{6 \left(\sin{\left(x \right)} + 4\right) \cos{\left(x \right)} \cos^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin^{2}{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}} - \frac{8 \sin{\left(x \right)} \cos{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}{\sin{\left(x - \frac{\cos{\left(x \right)}}{4} \right)}}}{32}$$