Detail solution
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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 sine is cosine:
The result of the chain rule is:
; to find :
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Rewrite the function to be differentiated:
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Apply the quotient rule, which is:
and .
To find :
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The derivative of sine is cosine:
To find :
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The derivative of cosine is negative sine:
Now plug in to the quotient rule:
The result is:
Now simplify:
The answer is:
The first derivative
[src]
2 / 2 \
sin (x)*\1 + tan (x)/ + 2*cos(x)*sin(x)*tan(x)
$$\left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}$$
The second derivative
[src]
/ / 2 2 \ 2 / 2 \ / 2 \ \
2*\- \sin (x) - cos (x)/*tan(x) + sin (x)*\1 + tan (x)/*tan(x) + 2*\1 + tan (x)/*cos(x)*sin(x)/
$$2 \left(- \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
The third derivative
[src]
/ / 2 \ / 2 2 \ 2 / 2 \ / 2 \ / 2 \ \
2*\- 3*\1 + tan (x)/*\sin (x) - cos (x)/ + sin (x)*\1 + tan (x)/*\1 + 3*tan (x)/ - 4*cos(x)*sin(x)*tan(x) + 6*\1 + tan (x)/*cos(x)*sin(x)*tan(x)/
$$2 \left(- 3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} - 4 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}\right)$$