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How to use it?
- Derivative of:
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Derivative of 7x
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Derivative of e^(2*cos(t)^(2)-2*sin(t)^(2))
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Derivative of 4/x^3
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Derivative of 4*sqrt(x)
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Identical expressions
- (x^ three + four)^tan(x/ two)
- (x cubed plus 4) to the power of tangent of (x divide by 2)
- (x to the power of three plus four) to the power of tangent of (x divide by two)
- (x3+4)tan(x/2)
- x3+4tanx/2
- (x³+4)^tan(x/2)
- (x to the power of 3+4) to the power of tan(x/2)
- x^3+4^tanx/2
- (x^3+4)^tan(x divide by 2)
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Similar expressions
- (x^3-4)^tan(x/2)
Derivative of (x^3+4)^tan(x/2)
The solution
You have entered
[src]
/x\
tan|-|
\2/
/ 3 \
\x + 4/
$$\left(x^{3} + 4\right)^{\tan{\left(\frac{x}{2} \right)}}$$
/ /x\\ | tan|-|| | \2/| d |/ 3 \ | --\\x + 4/ / dx
$$\frac{d}{d x} \left(x^{3} + 4\right)^{\tan{\left(\frac{x}{2} \right)}}$$
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
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Now simplify:
The answer is:
The first derivative
[src]
/x\
tan|-| // 2/x\\ 2 /x\\
\2/ || tan |-|| 3*x *tan|-||
/ 3 \ ||1 \2/| / 3 \ \2/|
\x + 4/ *||- + -------|*log\x + 4/ + -----------|
|\2 2 / 3 |
\ x + 4 /
$$\left(x^{3} + 4\right)^{\tan{\left(\frac{x}{2} \right)}} \left(\frac{3 x^{2} \tan{\left(\frac{x}{2} \right)}}{x^{3} + 4} + \left(\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}\right) \log{\left(x^{3} + 4 \right)}\right)$$
The second derivative
[src]
/ 2 \
|/ 2 /x\\ |
|| 6*x *tan|-|| |
/x\ ||/ 2/x\\ / 3\ \2/| |
tan|-| |||1 + tan |-||*log\4 + x / + -----------| / 2/x\\ / 3\ /x\ 4 /x\ 2 / 2/x\\ /x\|
\2/ ||\ \2// 3 | |1 + tan |-||*log\4 + x /*tan|-| 9*x *tan|-| 3*x *|1 + tan |-|| 6*x*tan|-||
/ 3\ |\ 4 + x / \ \2// \2/ \2/ \ \2// \2/|
\4 + x / *|------------------------------------------ + -------------------------------- - ----------- + ------------------ + ----------|
| 4 2 2 3 3 |
| / 3\ 4 + x 4 + x |
\ \4 + x / /
$$\left(x^{3} + 4\right)^{\tan{\left(\frac{x}{2} \right)}} \left(- \frac{9 x^{4} \tan{\left(\frac{x}{2} \right)}}{\left(x^{3} + 4\right)^{2}} + \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x^{3} + 4 \right)} \tan{\left(\frac{x}{2} \right)}}{2} + \frac{3 x^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{x^{3} + 4} + \frac{\left(\frac{6 x^{2} \tan{\left(\frac{x}{2} \right)}}{x^{3} + 4} + \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x^{3} + 4 \right)}\right)^{2}}{4} + \frac{6 x \tan{\left(\frac{x}{2} \right)}}{x^{3} + 4}\right)$$
The third derivative
[src]
/ 3 / 2 /x\\ / 4 /x\ 2 / 2/x\\ /x\\ \
|/ 2 /x\\ | 6*x *tan|-|| | 18*x *tan|-| 6*x *|1 + tan |-|| 12*x*tan|-|| |
|| 6*x *tan|-|| |/ 2/x\\ / 3\ \2/| |/ 2/x\\ / 3\ /x\ \2/ \ \2// \2/| |
/x\ ||/ 2/x\\ / 3\ \2/| 2 3*||1 + tan |-||*log\4 + x / + -----------|*||1 + tan |-||*log\4 + x /*tan|-| - ------------ + ------------------ + -----------| |
tan|-| |||1 + tan |-||*log\4 + x / + -----------| /x\ / 2/x\\ / 3\ |\ \2// 3 | |\ \2// \2/ 2 3 3 | 2/x\ / 2/x\\ / 3\ 3 /x\ / 2/x\\ 6 /x\ 4 / 2/x\\ 2 / 2/x\\ /x\|
\2/ ||\ \2// 3 | 6*tan|-| |1 + tan |-|| *log\4 + x / \ 4 + x / | / 3\ 4 + x 4 + x | tan |-|*|1 + tan |-||*log\4 + x / 54*x *tan|-| 9*x*|1 + tan |-|| 54*x *tan|-| 27*x *|1 + tan |-|| 9*x *|1 + tan |-||*tan|-||
/ 3\ |\ 4 + x / \2/ \ \2// \ \4 + x / / \2/ \ \2// \2/ \ \2// \2/ \ \2// \ \2// \2/|
\4 + x / *|------------------------------------------ + -------- + -------------------------- + -------------------------------------------------------------------------------------------------------------------------------- + --------------------------------- - ------------ + ----------------- + ------------ - ------------------- + -------------------------|
| 8 3 4 4 2 2 3 3 2 / 3\ |
| 4 + x / 3\ 4 + x / 3\ / 3\ 2*\4 + x / |
\ \4 + x / \4 + x / 2*\4 + x / /
$$\left(x^{3} + 4\right)^{\tan{\left(\frac{x}{2} \right)}} \left(\frac{54 x^{6} \tan{\left(\frac{x}{2} \right)}}{\left(x^{3} + 4\right)^{3}} + \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x^{3} + 4 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{27 x^{4} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{2 \left(x^{3} + 4\right)^{2}} + \frac{9 x^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2 \left(x^{3} + 4\right)} + \frac{\left(\frac{6 x^{2} \tan{\left(\frac{x}{2} \right)}}{x^{3} + 4} + \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x^{3} + 4 \right)}\right)^{3}}{8} + \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \log{\left(x^{3} + 4 \right)}}{4} - \frac{54 x^{3} \tan{\left(\frac{x}{2} \right)}}{\left(x^{3} + 4\right)^{2}} + \frac{3 \cdot \left(\frac{6 x^{2} \tan{\left(\frac{x}{2} \right)}}{x^{3} + 4} + \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x^{3} + 4 \right)}\right) \left(- \frac{18 x^{4} \tan{\left(\frac{x}{2} \right)}}{\left(x^{3} + 4\right)^{2}} + \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x^{3} + 4 \right)} \tan{\left(\frac{x}{2} \right)} + \frac{6 x^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{x^{3} + 4} + \frac{12 x \tan{\left(\frac{x}{2} \right)}}{x^{3} + 4}\right)}{4} + \frac{9 x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{x^{3} + 4} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{x^{3} + 4}\right)$$
The graph