Detail solution
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Apply the product rule:
; to find :
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; 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 sine is cosine:
The result of the chain rule is:
The result is:
Now simplify:
The answer is:
The first derivative
[src]
x 2 x
2 *sin (x)*log(2) + 2*2 *cos(x)*sin(x)
$$2^{x} \log{\left(2 \right)} \sin^{2}{\left(x \right)} + 2 \cdot 2^{x} \sin{\left(x \right)} \cos{\left(x \right)}$$
The second derivative
[src]
x / 2 2 2 2 \
2 *\- 2*sin (x) + 2*cos (x) + log (2)*sin (x) + 4*cos(x)*log(2)*sin(x)/
$$2^{x} \left(- 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} + 4 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)$$
The third derivative
[src]
x / 3 2 / 2 2 \ 2 \
2 *\log (2)*sin (x) - 8*cos(x)*sin(x) - 6*\sin (x) - cos (x)/*log(2) + 6*log (2)*cos(x)*sin(x)/
$$2^{x} \left(\log{\left(2 \right)}^{3} \sin^{2}{\left(x \right)} - 8 \sin{\left(x \right)} \cos{\left(x \right)} + 6 \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 6 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}\right)$$