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Chain Rule Differentiation Problem?
What is the derivative of 3^(7^(x^2)) ?
I tried logarithmic differentiation as well, but wasn't getting the right result, so I'm trying to use chain rule.
So far I have
u(x) = 3^7
g(x) = u^x
h(x) = (u^x)^2
I really want to learn how to do this question.... thanks guys, I'll def pick the best answer
2 Answers
- Anonymous9 years agoFavorite Answer
You need to use the Chain Rule twice.
Let y = 3^[7^(x^2)] and u = 7^(x^2). Hence y = 3^u
dy/du = 3^(u)ln(3) = 3^[7^(x^2)] ln(3)
To find du/dx use the Chain Rule for the first time thus:
Let Y = 7^(x^2) and let U = x^2. Then Y = 7^U. Hence:
dY/dU = 7U ln(7) = 7x^(2)ln(7) and DU/dx = 2x
Chain Rule: dY/dx = dY/dU/* dU/dx = 7x^(2)ln(7)*2x
Hence du/dx = 2x*7x^(2)ln(7)
Using the Chain Rule again: dy/dx = dy/du * du/dx.
dy/dx = 3^[7^(x^2)]ln(3) * 2x*7x^(2)ln(7)
dy/dx = 2x*3^[7^x^(2)]* 7^x^(2)*ln(3)*ln(7)
- Engr. RonaldLv 79 years ago
y = 3^(7^(x^2))
dy/dx = 3^[7^(x^2)]ln(3) d/dx 7^(x^2)ln(7) d/dx 2x
= 2x* 3^[7^(x^2)] * 7^(x^2)ln(3)ln(7) answer//
