Dy/dx of e^4square root of x

d/dx [(e^4) (x^.5)]

or do you mean
(d/dx e^4 ) x^.5
or do you mean
d/dx e^(4sqrtx)
???????

sorry i mean d/dx e^(4sqrtx)

To find the derivative dy/dx of e^(4√x), we can use the chain rule.

The chain rule states that if we have a function of the form f(g(x)), where f(u) is differentiable and g(x) is also differentiable, then the derivative of f(g(x)) is given by the product of the derivative of f(u) with respect to u, and the derivative of g(x) with respect to x.

Let's break down the function e^(4√x) into two parts:
- f(u) = e^u, where u = 4√x
- g(x) = 4√x

Now, let's find the derivative of each part separately:
- The derivative of f(u) = e^u with respect to u is simply e^u.
- The derivative of g(x) = 4√x with respect to x can be found by applying the chain rule again. We have g(x) = 4(x)^0.5, so the derivative is g'(x) = 4 * 0.5 * x^(-0.5) = 2/x^(0.5).

Now, apply the chain rule:
dy/dx = (df/du) * (dg/dx)
= e^u * 2/x^(0.5)
= e^(4√x) * 2/x^(0.5)

Therefore, the derivative dy/dx of e^(4√x) is e^(4√x) * 2/x^(0.5).