To determine

To find:

f' in terms of g'

Answer

f'x=g'(tan ⁡x) (sec2⁡x)2x

Explanation

Differentiate by using differentiation rules

1) Formula:

(i) Chain rule:

ddx(fgx=f'gx*g'(x)

(ii) Trigonometric rule:

ddxtan⁡x=sec2⁡x

(iii) Power rule:

ddxxn=n*xn-1

2) Given:

fx=g(tan⁡x)

3) Calculation:

Consider the function fx=g(tan⁡x)

Differentiate the function with respect to x,

f'x=ddx(g(tan⁡x))

By using chain rule,

f'x=g'(tan ⁡x) ddx(tan⁡x)

By using trigonometric rule and chain rule,

f'x=g'(tan ⁡x) (sec2⁡x)ddx(x)

By using power rule,

f'x=g'(tan ⁡x) (sec2⁡x)(12x)

Therefore,

f'x=g'(tan ⁡x) (sec2⁡x)2x

Conclusion:

f'x=g'(tan ⁡x) (sec2⁡x)2x