Detail solution
-
Apply the product rule:
; to find :
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Let .
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Apply the power rule: goes to
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Then, apply the chain rule. Multiply by :
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The derivative of is .
The result of the chain rule is:
; to find :
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The derivative of sine is cosine:
The result is:
Now simplify:
The answer is:
The first derivative
[src]
2
3 3*log (x)*sin(x)
log (x)*cos(x) + ----------------
x
$$\log{\left(x \right)}^{3} \cos{\left(x \right)} + \frac{3 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x}$$
The second derivative
[src]
/ 2 3*(-2 + log(x))*sin(x) 6*cos(x)*log(x)\
|- log (x)*sin(x) - ---------------------- + ---------------|*log(x)
| 2 x |
\ x /
$$\left(- \log{\left(x \right)}^{2} \sin{\left(x \right)} + \frac{6 \log{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{3 \left(\log{\left(x \right)} - 2\right) \sin{\left(x \right)}}{x^{2}}\right) \log{\left(x \right)}$$
The third derivative
[src]
2 / 2 \
3 9*log (x)*sin(x) 6*\1 + log (x) - 3*log(x)/*sin(x) 9*(-2 + log(x))*cos(x)*log(x)
- log (x)*cos(x) - ---------------- + --------------------------------- - -----------------------------
x 3 2
x x
$$- \log{\left(x \right)}^{3} \cos{\left(x \right)} - \frac{9 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x} - \frac{9 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{6 \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right) \sin{\left(x \right)}}{x^{3}}$$