APFIVE Chain Rule With Exponential and Trig Functions
Which of the following describes the correct application of the chain rule to find the derivative of the function $$y=\sin(e^{2x})$$?
A
Treat $$e^{2*x}$$ as a constant, differentiate $$\sin(u)$$ directly, and then multiply by the derivative of $$2*x$$
B
Differentiate $$\sin(e^{2*x})$$ and $$e^{2*x}$$ simultaneously, then multiply by 2 afterward, ignoring the nested structure
C
Differentiate $$e^{2*x}$$ first, then apply the derivative of sine directly without the proper chain rule for $$\sin(u)$$, and multiply by $$2*x$$
D
Identify the layers: outer $$\sin(u)$$, inner $$e^{2*x}$$, and innermost $$2*x$$; differentiate the outer function to get $$\cos(e^{2*x})$$; differentiate $$e^{2*x}$$ to get $$2*e^{2*x}$$; then multiply the derivatives
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