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How to use it?
- Derivative of:
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Derivative of x^7*e^x
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Derivative of x/3+3/x
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Derivative of 6^x
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Derivative of (x+3)^4
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Identical expressions
- x^ two (ln(x+sinx))(x+sinx)^x^ two
- x squared (ln(x plus sinus of x))(x plus sinus of x) to the power of x squared
- x to the power of two (ln(x plus sinus of x))(x plus sinus of x) to the power of x to the power of two
- x2(ln(x+sinx))(x+sinx)x2
- x2lnx+sinxx+sinxx2
- x²(ln(x+sinx))(x+sinx)^x²
- x to the power of 2(ln(x+sinx))(x+sinx) to the power of x to the power of 2
- x^2lnx+sinxx+sinx^x^2
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Similar expressions
- x^2(ln(x-sinx))(x+sinx)^x^2
- x^2(ln(x+sinx))(x-sinx)^x^2
Derivative of x^2(ln(x+sinx))(x+sinx)^x^2
The solution
You have entered
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/ 2\ 2 \x / x *log(x + sin(x))*(x + sin(x))
$$x^{2} \left(x + \sin{\left(x \right)}\right)^{x^{2}} \log{\left(x + \sin{\left(x \right)} \right)}$$
/ / 2\\ d | 2 \x /| --\x *log(x + sin(x))*(x + sin(x)) / dx
$$\frac{d}{d x} x^{2} \left(x + \sin{\left(x \right)}\right)^{x^{2}} \log{\left(x + \sin{\left(x \right)} \right)}$$
Detail solution
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Apply the product rule:
; to find :
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Apply the power rule: goes to
; to find :
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Let .
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The derivative of is .
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Then, apply the chain rule. Multiply by :
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Differentiate term by term:
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Apply the power rule: goes to
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The derivative of sine is cosine:
The result is:
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The result of the chain rule is:
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; to find :
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Don't know the steps in finding this derivative.
But the derivative is
The result is:
Now simplify:
The answer is:
The first derivative
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/ 2\
/ 2\ 2 \x / / 2\ / 2 \
\x / x *(x + sin(x)) *(1 + cos(x)) 2 \x / | x *(1 + cos(x))|
2*x*(x + sin(x)) *log(x + sin(x)) + -------------------------------- + x *(x + sin(x)) *|2*x*log(x + sin(x)) + ---------------|*log(x + sin(x))
x + sin(x) \ x + sin(x) /
$$x^{2} \left(x + \sin{\left(x \right)}\right)^{x^{2}} \left(\frac{x^{2} \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 x \log{\left(x + \sin{\left(x \right)} \right)}\right) \log{\left(x + \sin{\left(x \right)} \right)} + \frac{x^{2} \left(x + \sin{\left(x \right)}\right)^{x^{2}} \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 x \left(x + \sin{\left(x \right)}\right)^{x^{2}} \log{\left(x + \sin{\left(x \right)} \right)}$$
The second derivative
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/ / 2 \ \
| 2 |(1 + cos(x)) | 3 / x*(1 + cos(x))\|
/ 2\ | / 2 2 2 2 \ x *|------------- + sin(x)| 2*x *(1 + cos(x))*|2*log(x + sin(x)) + --------------||
\x / | 2 | 2 / x*(1 + cos(x))\ x *(1 + cos(x)) x *sin(x) 4*x*(1 + cos(x))| \ x + sin(x) / 4*x*(1 + cos(x)) 2 / x*(1 + cos(x))\ \ x + sin(x) /|
(x + sin(x)) *|2*log(x + sin(x)) + x *|2*log(x + sin(x)) + x *|2*log(x + sin(x)) + --------------| - ---------------- - ---------- + ----------------|*log(x + sin(x)) - --------------------------- + ---------------- + 4*x *|2*log(x + sin(x)) + --------------|*log(x + sin(x)) + ------------------------------------------------------|
| | \ x + sin(x) / 2 x + sin(x) x + sin(x) | x + sin(x) x + sin(x) \ x + sin(x) / x + sin(x) |
\ \ (x + sin(x)) / /
$$\left(x + \sin{\left(x \right)}\right)^{x^{2}} \cdot \left(\frac{2 x^{3} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 4 x^{2} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \log{\left(x + \sin{\left(x \right)} \right)} + x^{2} \left(x^{2} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right)^{2} - \frac{x^{2} \sin{\left(x \right)}}{x + \sin{\left(x \right)}} - \frac{x^{2} \left(\cos{\left(x \right)} + 1\right)^{2}}{\left(x + \sin{\left(x \right)}\right)^{2}} + \frac{4 x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \log{\left(x + \sin{\left(x \right)} \right)} - \frac{x^{2} \left(\sin{\left(x \right)} + \frac{\left(\cos{\left(x \right)} + 1\right)^{2}}{x + \sin{\left(x \right)}}\right)}{x + \sin{\left(x \right)}} + \frac{4 x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right)$$
The third derivative
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/ / 3 \ / 2 2 3 2 \ / 2 2 2 2 \ \
| 2 | 2*(1 + cos(x)) 3*(1 + cos(x))*sin(x)| | 2 6*x*(1 + cos(x)) 2*x *(1 + cos(x)) 3*x *(1 + cos(x))*sin(x) | / 2 \ / 2 \ 2 | 2 / x*(1 + cos(x))\ x *(1 + cos(x)) x *sin(x) 4*x*(1 + cos(x))| |
| x *|-cos(x) + --------------- + ---------------------| | 6 + 6*cos(x) - x *cos(x) - 6*x*sin(x) - ----------------- + ------------------ + ------------------------ | |(1 + cos(x)) | 3 / x*(1 + cos(x))\ |(1 + cos(x)) | 3*x *(1 + cos(x))*|2*log(x + sin(x)) + x *|2*log(x + sin(x)) + --------------| - ---------------- - ---------- + ----------------| 2 / x*(1 + cos(x))\|
/ 2\ | | 2 x + sin(x) | | 3 x + sin(x) 2 x + sin(x) / 2 2 2 \| 6*x*|------------- + sin(x)| / 2 2 2 2 \ 3*x *|2*log(x + sin(x)) + --------------|*|------------- + sin(x)| | \ x + sin(x) / 2 x + sin(x) x + sin(x) | 12*x *(1 + cos(x))*|2*log(x + sin(x)) + --------------||
\x / |6*(1 + cos(x)) \ (x + sin(x)) / 2 | 3 / x*(1 + cos(x))\ (x + sin(x)) / x*(1 + cos(x))\ | x *(1 + cos(x)) x *sin(x) 4*x*(1 + cos(x))|| \ x + sin(x) / / x*(1 + cos(x))\ | 2 / x*(1 + cos(x))\ x *(1 + cos(x)) x *sin(x) 4*x*(1 + cos(x))| \ x + sin(x) / \ x + sin(x) / \ (x + sin(x)) / \ x + sin(x) /|
(x + sin(x)) *|-------------- + ------------------------------------------------------ + x *|x *|2*log(x + sin(x)) + --------------| + --------------------------------------------------------------------------------------------------------- - 3*x*|2*log(x + sin(x)) + --------------|*|-2*log(x + sin(x)) + ---------------- + ---------- - ----------------||*log(x + sin(x)) - ---------------------------- + 6*x*|2*log(x + sin(x)) + --------------|*log(x + sin(x)) + 6*x*|2*log(x + sin(x)) + x *|2*log(x + sin(x)) + --------------| - ---------------- - ---------- + ----------------|*log(x + sin(x)) - ------------------------------------------------------------------ + ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------|
| x + sin(x) x + sin(x) | \ x + sin(x) / x + sin(x) \ x + sin(x) / | 2 x + sin(x) x + sin(x) || x + sin(x) \ x + sin(x) / | \ x + sin(x) / 2 x + sin(x) x + sin(x) | x + sin(x) x + sin(x) x + sin(x) |
\ \ \ (x + sin(x)) // \ (x + sin(x)) / /
$$\left(x + \sin{\left(x \right)}\right)^{x^{2}} \left(- \frac{3 x^{3} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \left(\sin{\left(x \right)} + \frac{\left(\cos{\left(x \right)} + 1\right)^{2}}{x + \sin{\left(x \right)}}\right)}{x + \sin{\left(x \right)}} + x^{2} \left(x^{3} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right)^{3} - 3 x \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \left(\frac{x^{2} \sin{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{x^{2} \left(\cos{\left(x \right)} + 1\right)^{2}}{\left(x + \sin{\left(x \right)}\right)^{2}} - \frac{4 x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} - 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) + \frac{- x^{2} \cos{\left(x \right)} + \frac{3 x^{2} \left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{2 x^{2} \left(\cos{\left(x \right)} + 1\right)^{3}}{\left(x + \sin{\left(x \right)}\right)^{2}} - 6 x \sin{\left(x \right)} - \frac{6 x \left(\cos{\left(x \right)} + 1\right)^{2}}{x + \sin{\left(x \right)}} + 6 \cos{\left(x \right)} + 6}{x + \sin{\left(x \right)}}\right) \log{\left(x + \sin{\left(x \right)} \right)} + \frac{12 x^{2} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + \frac{3 x^{2} \left(\cos{\left(x \right)} + 1\right) \left(x^{2} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right)^{2} - \frac{x^{2} \sin{\left(x \right)}}{x + \sin{\left(x \right)}} - \frac{x^{2} \left(\cos{\left(x \right)} + 1\right)^{2}}{\left(x + \sin{\left(x \right)}\right)^{2}} + \frac{4 x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right)}{x + \sin{\left(x \right)}} + \frac{x^{2} \left(- \cos{\left(x \right)} + \frac{3 \left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{2 \left(\cos{\left(x \right)} + 1\right)^{3}}{\left(x + \sin{\left(x \right)}\right)^{2}}\right)}{x + \sin{\left(x \right)}} + 6 x \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \log{\left(x + \sin{\left(x \right)} \right)} + 6 x \left(x^{2} \left(\frac{x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right)^{2} - \frac{x^{2} \sin{\left(x \right)}}{x + \sin{\left(x \right)}} - \frac{x^{2} \left(\cos{\left(x \right)} + 1\right)^{2}}{\left(x + \sin{\left(x \right)}\right)^{2}} + \frac{4 x \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}} + 2 \log{\left(x + \sin{\left(x \right)} \right)}\right) \log{\left(x + \sin{\left(x \right)} \right)} - \frac{6 x \left(\sin{\left(x \right)} + \frac{\left(\cos{\left(x \right)} + 1\right)^{2}}{x + \sin{\left(x \right)}}\right)}{x + \sin{\left(x \right)}} + \frac{6 \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}}\right)$$
The graph