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- Derivative of a implicit defined function
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- Partial derivative of the function
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How to use it?
- Derivative of:
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Derivative of x^3*lnx
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Derivative of sin(3)^(2)*x
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Derivative of (8*x-15)^5
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Derivative of (3*x-2)^7
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Identical expressions
- (one)/((cos^ two)*x)
- (1) divide by (( co sinus of e of squared ) multiply by x)
- (one) divide by (( co sinus of e of to the power of two) multiply by x)
- (1)/((cos2)*x)
- 1/cos2*x
- (1)/((cos²)*x)
- (1)/((cos to the power of 2)*x)
- (1)/((cos^2)x)
- (1)/((cos2)x)
- 1/cos2x
- 1/cos^2x
- (1) divide by ((cos^2)*x)
Derivative of (1)/((cos^2)*x)
The solution
You have entered
[src]
1 --------- 2 cos (x)*x
$$\frac{1}{x \cos^{2}{\left(x \right)}}$$
1/(cos(x)^2*x)
Detail solution
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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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Apply the product rule:
; to find :
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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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; to find :
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Apply the power rule: goes to
The result is:
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The result of the chain rule is:
Now simplify:
The answer is:
The first derivative
[src]
1 / 2 \
---------*\- cos (x) + 2*x*cos(x)*sin(x)/
2
x*cos (x)
-----------------------------------------
2
x*cos (x)
$$\frac{\frac{1}{x \cos^{2}{\left(x \right)}} \left(2 x \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{x \cos^{2}{\left(x \right)}}$$
The second derivative
[src]
/ 2 2 \
/ 1 2*sin(x)\ -cos(x) + 2*x*sin(x) 2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/ 2*(-cos(x) + 2*x*sin(x))*sin(x)
|- - + --------|*(-cos(x) + 2*x*sin(x)) - -------------------- + ------------------------------------------- + -------------------------------
\ x cos(x) / x cos(x) cos(x)
----------------------------------------------------------------------------------------------------------------------------------------------
2 3
x *cos (x)
$$\frac{\left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{1}{x}\right) + \frac{2 \left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{2 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\cos{\left(x \right)}} - \frac{2 x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}}{x^{2} \cos^{3}{\left(x \right)}}$$
The third derivative
[src]
/ 1 2*sin(x)\ / 1 2*sin(x)\ / 2 2 \ / 1 2*sin(x)\
/ 2 2 \ / 2 \ |- - + --------|*(-cos(x) + 2*x*sin(x)) / 2 2 \ 2*|- - + --------|*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/ 2 / 2 2 \ 2*|- - + --------|*(-cos(x) + 2*x*sin(x))*sin(x)
2*\- 3*cos (x) + 3*sin (x) + 4*x*cos(x)*sin(x)/ | 1 3*sin (x) 2*sin(x)| 3*(-cos(x) + 2*x*sin(x)) \ x cos(x) / 6*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/ \ x cos(x) / 10*sin (x)*(-cos(x) + 2*x*sin(x)) 12*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*sin(x) 8*(-cos(x) + 2*x*sin(x))*sin(x) \ x cos(x) /
-2*cos(x) - ----------------------------------------------- + 2*(-cos(x) + 2*x*sin(x))*|1 + -- + --------- - --------| + ------------------------ + 4*x*sin(x) - --------------------------------------- - ------------------------------------------- + ------------------------------------------------------------ + --------------------------------- + --------------------------------------------------- - ------------------------------- + ------------------------------------------------
cos(x) | 2 2 x*cos(x)| 2 x x*cos(x) cos(x) 2 2 x*cos(x) cos(x)
\ x cos (x) / x cos (x) cos (x)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 3
x *cos (x)
$$\frac{4 x \sin{\left(x \right)} + \frac{2 \left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{1}{x}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 - \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{1}{x^{2}}\right) + \frac{10 \left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{1}{x}\right) \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{12 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2 \left(4 x \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right)}{\cos{\left(x \right)}} - 2 \cos{\left(x \right)} - \frac{\left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{1}{x}\right)}{x} - \frac{8 \left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{6 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}} + \frac{3 \left(2 x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x^{2}}}{x^{2} \cos^{3}{\left(x \right)}}$$