Question

Compute d/dx[f(g(x))]

-where f(x)=x^4-x^2
-and g(x)=x^2-7


(Note: I tried 5 times but I still got this wrong. Plz help)

Answer 1

No implicit derivatives here -- just the chain rule and lots of tedious algebra...

df/dx = df/dg * dg/dx
= (4g^3 - g^2)(2x)
= (4(x^2-7)^3 - 2(x^2-7))(2x)
= 8x^7 - 168x^5 + 1172x^3 - 2716x

or, going directly,

f(g) = g^4-g^2
= (x^2-7)^4 - (x^2-7)^2
or,
x^8 - 28x^6 + 293x^4 - 1358x^2 + 2352

df/dx = 4(x^2-7)^3(2x) - 2(x^2-7)(2x)
= 8x^7 - 168x^5 + 1172x^3 - 2716x