y=e^(5e^x)

differentiate first and second derivative

u=5e^x

u' = 5e^x

y = e^u
y' = e^u u' = e^(5e^x) * 5e^x
or, y' = 5e^(x+5e^x)
y" = 5(1+5e^x)e^(x+5e^x)

To find the first derivative of the function y = e^(5e^x), we can use the chain rule.

Step 1: Identify the composite function within the outer function.
In this case, the composite function is 5e^x.

Step 2: Differentiate the outer function.
The derivative of e^u with respect to u is du/dx * e^u, where u represents the composite function. In our case, u is 5e^x.

Step 3: Differentiate the composite function.
The derivative of 5e^x is 5e^x * d(x)/dx, which simplifies to 5e^x.

Step 4: Multiply the derivative of the outer function with the derivative of the composite function to obtain the first derivative.
The first derivative of y = e^(5e^x) is dy/dx = (5e^x) * e^(5e^x).

To find the second derivative, we need to differentiate the first derivative obtained above with respect to x. Let's go through the steps:

Step 1: Identify the composite function within the outer function.
In this case, the composite function is 5e^x.

Step 2: Differentiate the outer function.
The derivative of e^u with respect to u is du/dx * e^u, where u represents the composite function. In our case, u is 5e^x.

Step 3: Differentiate the composite function.
The derivative of 5e^x is 5e^x * d(x)/dx, which simplifies to 5e^x.

Step 4: Multiply the derivative of the outer function with the derivative of the composite function to obtain the first derivative.
The first derivative of y = (5e^x) * e^(5e^x) is dy/dx = 5e^x * e^(5e^x).

Step 5: Differentiate the first derivative obtained in Step 4.
The second derivative of y = 5e^x * e^(5e^x) is given by the product rule: d[(5e^x * e^(5e^x))] / dx = (d(5e^x)/dx) * e^(5e^x) + (5e^x) * d[e^(5e^x)]/dx.

Simplifying the above expression, we have:
dy^2/dx^2 = 5e^x * e^(5e^x) + (5e^x) * (d(5e^x)/dx) * e^(5e^x).

Applying the chain rule again, we find that (d(5e^x)/dx) = 5e^x.

Therefore, the second derivative is:
dy^2/dx^2 = 5e^x * e^(5e^x) + (5e^x) * (5e^x) * e^(5e^x).

Simplifying further gives:
dy^2/dx^2 = 5e^x * e^(5e^x) + 25e^(2x) * e^(5e^x).

Hence, the second derivative is given by dy^2/dx^2 = 5e^x * e^(5e^x) + 25e^(2x) * e^(5e^x).