Use the Chain Rule to find 𝑑𝑦 𝑑𝑥 in terms of 𝑥 and simplify the result

Question

𝑦 = 𝑒^−𝑢 and 𝑢 = 𝑒^−𝑥

Answer 1

we use chain rule when taking the derivative of a function that depends on a function of x (rather than just x itself). In other words, if h(g(x)), then its derivative (from chain rule) is

dh/dx = dh/dg*dg/dx (or the derivative of the "outside" times the derivative of the "inside")

back to our problem,

u = e^-x

du/dx = d/dx (e^-x) = -e^-x

y = e^-u(x)

dy/dx = d/dx (e^-u(x)) = -e^-u(x) * du/dx (plug in from above and you're done)

Answer 2

𝑦 = 𝑒^−𝑢 and 𝑢 = 𝑒^−𝑥

For the first equation:

ln (y) = -u

Then substitute the value of u in terms of x.

(A) ln(y) = -e^-x

(B) y = e^{-e^x}

Get the derivative of both sides of the equation (A) with respect to x.

(1/y)(dy/dx) = -e^-x (-1) (This is where the chain rule happens)

dy/dx = ye^-x

Substitute y in terms of x:

dy/dx = e^{-e^x}e^-x

dy/dx = e^{-e^x - x}