# We now have

d
dx
[x5 + y8] = 5x4 + 8y7y' =
d
dx
[9] = 0.

Rearranging this, we get
8y7y' = ??????????

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A table of values for f, g, f ', and g' is given.
x f(x) g(x) f '(x) g'(x)
1 3 2 4 6
2 1 8 5 7
3 7 2 7 9
(a) If h(x) = f(g(x)), find h'(3).
h'(3) =

(b) If
H(x) = g(f(x)), find H'(1).
H'(1) =

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Let f and g be the functions in the table below.
x f(x) g(x) f '(x) g'(x)
1 3 2 4 6
2 1 3 5 7
3 2 1 7 9
(a) If F(x) = f(f(x)), find F '(3).
F '(3) =

(b) If G(x) = g(g(x)), find G'(2).
G'(2) =

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If h(x) = Squareroot 4 + 3f(x), where f(3) = 4 and f '(3) = 3, find h'(3).
h'(3) =
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Write the composite function in the form
f(g(x)). [Identify the inner function
u = g(x and the outer function y = f(u).]
y = e^ 5 Squareroot x
(g(x), f(u)) =
Find the derivative dy/dx.

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1. d/dx (x^5 + y^8) is indeed 5x^4 + 8y^7 y'
Usually this is done in conjunction with implicit derivatives, where you have some equation like

x^5 + y^8 = 7
and you wind up with
y' = -5x^4/8y^7
which you then have to evaluate at some point.

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h(x) = f(g(x))
h' = f'(g(x)) * g', so
h'(3) = f'(g(3)) g'(3) = f'(2) g'(3) = 5*9 = 45

b is the same thing
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This is just the same, but using different functions

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h(x) = √(4 + 3f(x))
h' = 3f'/2√(4+3f)
h'(3) = 3*3/√(4+3*4) = 9/√16 = 9/4

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If
f(x) = e^x
g(x) = 5√x, then
y = f(u) = e^u = e^(5√x)

y' = 5/(2√x) * e^5√x

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