Detail solution
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Apply the product rule:
; 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:
; to find :
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The derivative of sine is cosine:
The result is:
Now simplify:
The answer is:
The first derivative
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/ 2 \
\1 + tan (x)/*sin(x) + cos(x)*tan(x)
$$\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}$$
The second derivative
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/ 2 \ / 2 \
-sin(x)*tan(x) + 2*\1 + tan (x)/*cos(x) + 2*\1 + tan (x)/*sin(x)*tan(x)
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \tan{\left(x \right)}$$
The third derivative
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/ 2 \ / 2 \ / 2 \ / 2 \
-cos(x)*tan(x) - 3*\1 + tan (x)/*sin(x) + 2*\1 + tan (x)/*\1 + 3*tan (x)/*sin(x) + 6*\1 + tan (x)/*cos(x)*tan(x)
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} - \cos{\left(x \right)} \tan{\left(x \right)}$$