Composite Functions: Is it always true that the derivative of their composition is the composition of their derivatives?

Question

Given two differentiable functions f and g, their composition is y=f(g(x)).

Is it always true that the derivative of their composition is the composition of

their derivatives? If so, explain why. If not, then give a clear

counter-example and show that the statement doesn’t apply.

Answer 1

Let Take 2 function f(x) = x² and g(x) = 2x + 1.

Then composition (fοg)(x) = f(g(x)) = (2x + 1)² = 4x² + 4x + 1, d/dx((fοg)(x)) = 8x + 4

Now df(x)/dx = 2x, dg(x)/dx = 2. Then compostion of these derivatives is: 8·2 + 4 = 20.

So, no derivative of composition of 2 functions not always equal composition of derivatives