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How to use it?
- Derivative of:
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Derivative of (-2)/x^3
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Derivative of ln(x^2-4x)
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Derivative of e^(3/x)
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Derivative of (2*x+1)*(x^2+3*x-1)
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Identical expressions
- (one +tgx)^ctgx
- (1 plus tgx) to the power of ctgx
- (one plus tgx) to the power of ctgx
- (1+tgx)ctgx
- 1+tgxctgx
- 1+tgx^ctgx
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Similar expressions
- (1-tgx)^ctgx
Derivative of (1+tgx)^ctgx
The solution
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[src]
cot(x) (1 + tan(x))
$$\left(\tan{\left(x \right)} + 1\right)^{\cot{\left(x \right)}}$$
(1 + tan(x))^cot(x)
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]
/ / 2 \ \
cot(x) |/ 2 \ \1 + tan (x)/*cot(x)|
(1 + tan(x)) *|\-1 - cot (x)/*log(1 + tan(x)) + --------------------|
\ 1 + tan(x) /
$$\left(\left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(\tan{\left(x \right)} + 1 \right)} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)} + 1}\right) \left(\tan{\left(x \right)} + 1\right)^{\cot{\left(x \right)}}$$
The second derivative
[src]
/ 2 2 \
|/ / 2 \ \ / 2 \ / 2 \ / 2 \ / 2 \ |
cot(x) ||/ 2 \ \1 + tan (x)/*cot(x)| \1 + tan (x)/ *cot(x) 2*\1 + cot (x)/*\1 + tan (x)/ / 2 \ 2*\1 + tan (x)/*cot(x)*tan(x)|
(1 + tan(x)) *||\1 + cot (x)/*log(1 + tan(x)) - --------------------| - --------------------- - ----------------------------- + 2*\1 + cot (x)/*cot(x)*log(1 + tan(x)) + -----------------------------|
|\ 1 + tan(x) / 2 1 + tan(x) 1 + tan(x) |
\ (1 + tan(x)) /
$$\left(\tan{\left(x \right)} + 1\right)^{\cot{\left(x \right)}} \left(\left(\left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} + 1 \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)} + 1}\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} + 1 \right)} \cot{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)} + 1} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}\right)$$
The third derivative
[src]
/ 3 / 2 \ 2 3 2 2 \
| / / 2 \ \ / / 2 \ \ | / 2 \ / 2 \ / 2 \ / 2 \ | 2 / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ 2 / 2 \ / 2 \ / 2 \ |
cot(x) | |/ 2 \ \1 + tan (x)/*cot(x)| |/ 2 \ \1 + tan (x)/*cot(x)| | \1 + tan (x)/ *cot(x) 2*\1 + cot (x)/*\1 + tan (x)/ / 2 \ 2*\1 + tan (x)/*cot(x)*tan(x)| / 2 \ 2 / 2 \ 2*\1 + tan (x)/ *cot(x) 2*\1 + tan (x)/ *cot(x) 3*\1 + tan (x)/ *\1 + cot (x)/ 6*\1 + tan (x)/ *cot(x)*tan(x) 6*\1 + cot (x)/*\1 + tan (x)/*tan(x) 4*tan (x)*\1 + tan (x)/*cot(x) 6*\1 + cot (x)/*\1 + tan (x)/*cot(x)|
(1 + tan(x)) *|- |\1 + cot (x)/*log(1 + tan(x)) - --------------------| - 3*|\1 + cot (x)/*log(1 + tan(x)) - --------------------|*|- --------------------- - ----------------------------- + 2*\1 + cot (x)/*cot(x)*log(1 + tan(x)) + -----------------------------| - 2*\1 + cot (x)/ *log(1 + tan(x)) - 4*cot (x)*\1 + cot (x)/*log(1 + tan(x)) + ----------------------- + ----------------------- + ------------------------------ - ------------------------------ - ------------------------------------ + ------------------------------ + ------------------------------------|
| \ 1 + tan(x) / \ 1 + tan(x) / | 2 1 + tan(x) 1 + tan(x) | 1 + tan(x) 3 2 2 1 + tan(x) 1 + tan(x) 1 + tan(x) |
\ \ (1 + tan(x)) / (1 + tan(x)) (1 + tan(x)) (1 + tan(x)) /
$$\left(\tan{\left(x \right)} + 1\right)^{\cot{\left(x \right)}} \left(- \left(\left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} + 1 \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)} + 1}\right)^{3} - 3 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} + 1 \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)} + 1}\right) \left(2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} + 1 \right)} \cot{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)} + 1} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}\right) - 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} + 1 \right)} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} + 1 \right)} \cot^{2}{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} \cot{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} \cot{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \cot{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}}\right)$$