Question

Find the derivative of Sin(x)Tan(x) w.r.t e^x Tan(x)

(a) \(\frac{Sin(x)(1+Sec^4 (x))}{e^x (1+Tan^2 (x)+Tan(x))}\)

(b) \(\frac{Sin(x)(1+Sec^2 (x))}{e^x (1+Tan^4 (x)+Tan(x))}\)

(c) \(\frac{Sin(x)(1+Sec^2 (x))}{e^x (1+Tan^2 (x)+Tan(x))}\)

(d) \(\frac{Sin(x)(1+Sec^2 (x))}{e^x (2+Tan^2 (x)+Tan(x))}\)

I had been asked this question in exam.

This interesting question is from Limits and Derivatives of Several Variables topic in section Partial Differentiation of Engineering Mathematics

Answer 1

Correct option is (c) \(\frac{Sin(x)(1+Sec^2 (x))}{e^x (1+Tan^2 (x)+Tan(x))}\)

The best I can explain: \(\frac{dSin(x)Tan(x)}{de^x Tan(x)} = \frac{dSin(x)Tan(x))}{dx} \frac{dx}{de^x Tan(x)}\)

=\(\frac{dSin(x)Tan(x)}{dx} \frac{1}{\frac{de^x Tan(x)}{dx}}\)

=\(\frac{[Sin(x)Sec^2 (x)+Sin(x)]}{e^x (Sec^2 (x)+Tan(x))}\)

=\(\frac{Sin(x)(1+Sec^2 (x))}{e^x (1+Tan^2 (x)+Tan(x))}\)