Mister Exam
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- Derivative of a implicit defined function
- Derivative of Parametric Function
- 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 y/2
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Derivative of (x+5)/(x-1)
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Derivative of x+9
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Derivative of (x-4)
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Identical expressions
- (cos(5x))^e^x
- ( co sinus of e of (5x)) to the power of e to the power of x
- (cos(5x))ex
- cos5xex
- cos5x^e^x
Derivative of (cos(5x))^e^x
The solution
You have entered
[src]
/ x\
\e /
(cos(5*x))
$$\cos^{e^{x}}{\left(5 x \right)}$$
/ / x\\ d | \e /| --\(cos(5*x)) / dx
$$\frac{d}{d x} \cos^{e^{x}}{\left(5 x \right)}$$
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
[src]
/ x\ / x \
\e / | x 5*e *sin(5*x)|
(cos(5*x)) *|e *log(cos(5*x)) - -------------|
\ cos(5*x) /
$$\left(e^{x} \log{\left(\cos{\left(5 x \right)} \right)} - \frac{5 e^{x} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}\right) \cos^{e^{x}}{\left(5 x \right)}$$
The second derivative
[src]
/ x\ / 2 2 \
\e / | / 5*sin(5*x) \ x 25*sin (5*x) 10*sin(5*x) | x
(cos(5*x)) *|-25 + |- ---------- + log(cos(5*x))| *e - ------------ - ----------- + log(cos(5*x))|*e
| \ cos(5*x) / 2 cos(5*x) |
\ cos (5*x) /
$$\left(\left(\log{\left(\cos{\left(5 x \right)} \right)} - \frac{5 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}\right)^{2} e^{x} + \log{\left(\cos{\left(5 x \right)} \right)} - \frac{25 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} - \frac{10 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - 25\right) e^{x} \cos^{e^{x}}{\left(5 x \right)}$$
The third derivative
[src]
/ x\ / 3 3 2 / 2 \ \
\e / | / 5*sin(5*x) \ 2*x 265*sin(5*x) 250*sin (5*x) 75*sin (5*x) / 5*sin(5*x) \ | 10*sin(5*x) 25*sin (5*x)| x | x
(cos(5*x)) *|-75 + |- ---------- + log(cos(5*x))| *e - ------------ - ------------- - ------------ - 3*|- ---------- + log(cos(5*x))|*|25 - log(cos(5*x)) + ----------- + ------------|*e + log(cos(5*x))|*e
| \ cos(5*x) / cos(5*x) 3 2 \ cos(5*x) / | cos(5*x) 2 | |
\ cos (5*x) cos (5*x) \ cos (5*x) / /
$$\left(\left(\log{\left(\cos{\left(5 x \right)} \right)} - \frac{5 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}\right)^{3} e^{2 x} - 3 \left(\log{\left(\cos{\left(5 x \right)} \right)} - \frac{5 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}\right) \left(- \log{\left(\cos{\left(5 x \right)} \right)} + \frac{25 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + \frac{10 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} + 25\right) e^{x} + \log{\left(\cos{\left(5 x \right)} \right)} - \frac{250 \sin^{3}{\left(5 x \right)}}{\cos^{3}{\left(5 x \right)}} - \frac{75 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} - \frac{265 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - 75\right) e^{x} \cos^{e^{x}}{\left(5 x \right)}$$
The graph