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+3)^4
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Derivative of (x^2-x)*(x^3+x)
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Derivative of (x-2)^4
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Derivative of (x^2-1)*(x^4+2)
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Identical expressions
- ln*((one +sinx)/cosx)
- ln multiply by ((1 plus sinus of x) divide by co sinus of e of x)
- ln multiply by ((one plus sinus of x) divide by co sinus of e of x)
- ln((1+sinx)/cosx)
- ln1+sinx/cosx
- ln*((1+sinx) divide by cosx)
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Similar expressions
- sinx/cos^2x+ln1+(sinx)/(cosx)
- (sinx/cos^2x)+ln*(1+sinx/cosx)
- ln*((1-sinx)/cosx)
- (sin(x)+cos(x))+(ln((1+sin(x))/cos(x)))
- y=ln1+sinx/cosx
Derivative of ln*((1+sinx)/cosx)
The solution
You have entered
[src]
/1 + sin(x)\ log|----------| \ cos(x) /
$$\log{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)}} \right)}$$
log((1 + sin(x))/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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Differentiate term by term:
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The derivative of the constant is zero.
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The derivative of sine is cosine:
The result is:
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To find :
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The derivative of cosine is negative sine:
Now plug in to the quotient rule:
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The result of the chain rule is:
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Now simplify:
The answer is:
The first derivative
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/ (1 + sin(x))*sin(x)\
|1 + -------------------|*cos(x)
| 2 |
\ cos (x) /
--------------------------------
1 + sin(x)
$$\frac{\left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)} + 1}$$
The second derivative
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2 / (1 + sin(x))*sin(x)\
cos (x)*|1 + -------------------|
| 2 | 2
/ (1 + sin(x))*sin(x)\ \ cos (x) / 2*sin (x)*(1 + sin(x))
1 + 2*sin(x) - |1 + -------------------|*sin(x) - --------------------------------- + ----------------------
| 2 | 1 + sin(x) 2
\ cos (x) / cos (x)
------------------------------------------------------------------------------------------------------------
1 + sin(x)
$$\frac{- \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)} - \frac{\left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + 1}{\sin{\left(x \right)} + 1}$$
The third derivative
[src]
/ 2 \ / 2 \
| 2*sin (x)*(1 + sin(x))| | 2*sin (x)*(1 + sin(x))| 3 / (1 + sin(x))*sin(x)\ / (1 + sin(x))*sin(x)\
2*|1 + 2*sin(x) + ----------------------|*cos(x) 2*|1 + 2*sin(x) + ----------------------|*sin(x) 2*cos (x)*|1 + -------------------| 3*|1 + -------------------|*cos(x)*sin(x)
/ 2 3 \ | 2 | | 2 | | 2 | | 2 |
| 3*sin (x) 5*(1 + sin(x))*sin(x) 6*sin (x)*(1 + sin(x))| / (1 + sin(x))*sin(x)\ \ cos (x) / \ cos (x) / \ cos (x) / \ cos (x) /
|2 + --------- + --------------------- + ----------------------|*cos(x) - |1 + -------------------|*cos(x) - ------------------------------------------------ - ------------------------------------------------ + ----------------------------------- + -----------------------------------------
| 2 2 4 | | 2 | 1 + sin(x) cos(x) 2 1 + sin(x)
\ cos (x) cos (x) cos (x) / \ cos (x) / (1 + sin(x))
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 + sin(x)
$$\frac{- \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)} + \frac{3 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{2 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos^{3}{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} - \frac{2 \left(\frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \left(\frac{6 \left(\sin{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{5 \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \cos{\left(x \right)} - \frac{2 \left(\frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)} + 1}}{\sin{\left(x \right)} + 1}$$