Find dy/dx given the following functions

Question

Given y=f(u) and u=g(x), find dy/dx if f'(g(x)g'(x) for the following functions. y=5√u, u= cos x

Answer 1

Given y = f(u) and u = g(x),

that means that y = f(g(x)).

In other words, the function y is the composition of the functions f and g.

Therefore, by the Chain Rule,

dy/dx = f'(g(x))g'(x).

Note that you forgot to write one of the right parentheses in the expression f'(g(x))g'(x).

Now let's say what the derivatives of f and g individually are.

y = f(u) = 5√u = 5u^1/2,

so f'(u) = (5/2)u^-1/2 = 5/(2√u)

u = g(x) = cos x,

so g'(x) = -sin x

Therefore dy/dx = f'(g(x))g'(x) = (5/(2√(cos x)))(-sin x)

Let's write this more legibly as (-5sin x) / (2√(cos x))

Thanks.