Write the composite function in the form f(g(x)).
[Identify the inner function u = g(x) and the outer function y = f(u).]
(Use non-identity functions for f(u) and g(x).)
y=e^9 √x
(f(u), g(x)) =
Find the derivative dy/dx
dy/dx=
y = e^(9√x)
y = 9/(2√x) e^(9√x)
(f(g(x)), g(x)) = (e^u, √x)
Here, the inner function is u = g(x) = √x and the outer function is y = f(u) = e^u.
To find the derivative dy/dx, we can use the chain rule:
dy/dx = dy/du * du/dx
We know that dy/du = e^u and du/dx = 1/(2√x) (since u = √x).
Therefore,
dy/dx = dy/du * du/dx = e^u * 1/(2√x) = e^√x / (2√x)
So, the derivative of y = e^9 √x is dy/dx = e^√x / (2√x).