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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
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Derivative of f(x)=4x^2
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Derivative of (sin(x))^2/cos(2*x)
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Derivative of 7*x+5
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Derivative of y=cosx/x
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Identical expressions
- y=(x^ two + three)^ctgx
- y equally (x squared plus 3) to the power of ctgx
- y equally (x to the power of two plus three) to the power of ctgx
- y=(x2+3)ctgx
- y=x2+3ctgx
- y=(x²+3)^ctgx
- y=(x to the power of 2+3) to the power of ctgx
- y=x^2+3^ctgx
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Similar expressions
- y=(x^2-3)^ctgx
Derivative of y=(x^2+3)^ctgx
The solution
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cot(x) / 2 \ \x + 3/
$$\left(x^{2} + 3\right)^{\cot{\left(x \right)}}$$
(x^2 + 3)^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
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cot(x)
/ 2 \ // 2 \ / 2 \ 2*x*cot(x)\
\x + 3/ *|\-1 - cot (x)/*log\x + 3/ + ----------|
| 2 |
\ x + 3 /
$$\left(x^{2} + 3\right)^{\cot{\left(x \right)}} \left(\frac{2 x \cot{\left(x \right)}}{x^{2} + 3} + \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x^{2} + 3 \right)}\right)$$
The second derivative
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cot(x) / 2 / 2 \ 2 \
/ 2\ |/ / 2 \ / 2\ 2*x*cot(x)\ 2*cot(x) 4*x*\1 + cot (x)/ 4*x *cot(x) / 2 \ / 2\|
\3 + x / *||- \1 + cot (x)/*log\3 + x / + ----------| + -------- - ----------------- - ----------- + 2*\1 + cot (x)/*cot(x)*log\3 + x /|
|| 2 | 2 2 2 |
|\ 3 + x / 3 + x 3 + x / 2\ |
\ \3 + x / /
$$\left(x^{2} + 3\right)^{\cot{\left(x \right)}} \left(- \frac{4 x^{2} \cot{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} - \frac{4 x \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} + 3} + \left(\frac{2 x \cot{\left(x \right)}}{x^{2} + 3} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{2} + 3 \right)}\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{2} + 3 \right)} \cot{\left(x \right)} + \frac{2 \cot{\left(x \right)}}{x^{2} + 3}\right)$$
The third derivative
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cot(x) / 3 / 2 \ / / 2 \ 2 \ 2 2 / 2 \ 3 / 2 \ \
/ 2\ |/ / 2 \ / 2\ 2*x*cot(x)\ 6*\1 + cot (x)/ / / 2 \ / 2\ 2*x*cot(x)\ | cot(x) / 2 \ / 2\ 2*x*\1 + cot (x)/ 2*x *cot(x)| / 2 \ / 2\ 12*x*cot(x) 2 / 2 \ / 2\ 12*x *\1 + cot (x)/ 16*x *cot(x) 12*x*\1 + cot (x)/*cot(x)|
\3 + x / *||- \1 + cot (x)/*log\3 + x / + ----------| - --------------- - 6*|- \1 + cot (x)/*log\3 + x / + ----------|*|- ------ - \1 + cot (x)/*cot(x)*log\3 + x / + ----------------- + -----------| - 2*\1 + cot (x)/ *log\3 + x / - ----------- - 4*cot (x)*\1 + cot (x)/*log\3 + x / + ------------------- + ------------ + -------------------------|
|| 2 | 2 | 2 | | 2 2 2 | 2 2 3 2 |
|\ 3 + x / 3 + x \ 3 + x / | 3 + x 3 + x / 2\ | / 2\ / 2\ / 2\ 3 + x |
\ \ \3 + x / / \3 + x / \3 + x / \3 + x / /
$$\left(x^{2} + 3\right)^{\cot{\left(x \right)}} \left(\frac{16 x^{3} \cot{\left(x \right)}}{\left(x^{2} + 3\right)^{3}} + \frac{12 x^{2} \left(\cot^{2}{\left(x \right)} + 1\right)}{\left(x^{2} + 3\right)^{2}} + \frac{12 x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x^{2} + 3} - \frac{12 x \cot{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} + \left(\frac{2 x \cot{\left(x \right)}}{x^{2} + 3} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{2} + 3 \right)}\right)^{3} - 6 \left(\frac{2 x \cot{\left(x \right)}}{x^{2} + 3} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{2} + 3 \right)}\right) \left(\frac{2 x^{2} \cot{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} + \frac{2 x \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} + 3} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{2} + 3 \right)} \cot{\left(x \right)} - \frac{\cot{\left(x \right)}}{x^{2} + 3}\right) - 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x^{2} + 3 \right)} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x^{2} + 3 \right)} \cot^{2}{\left(x \right)} - \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} + 3}\right)$$
The graph