Michael J. answered • 10/15/15
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Mastery of Limits, Derivatives, and Integration Techniques
f(x) + x2[f(x)]3 = 10
f(1) = 2 means that when x=1 , f(x)=2. Plug in these values into the first equation.
2 + 12(2)3 = 10
2 + 8 = 10
10 = 10
The statement true. So we have to evaluate the derivative of f(x) using the point (1, 2) using implicit differentiation. This differentiation method uses the chain rule.
Let f(x) = y
y + x2y3 = 10
Derive both sides of the equation.
y' + 2xy3 + 3x2y2y' = 0
Isolate all the y' terms on side of equation. All the other terms on the opposite side of equation.
y' + 3x2y2y' = -2xy3
Factor out a y'.
y' (1 + 3x2y2) = -2xy3
f'(x) = (-2xy3) / (1 + 3x2y2)
Evaluate f'(x) at x=1.
f'(1) = (-2(23)) / (1 + 3(22))
f'(1) = -16 / (1 + 12)
f'(1) = -16 / 13
Hanna N.
you're using the point you determined earlier, (1,2), and since 1 to the power of anything is 1, they just multiplied it in, you don't need to focus on it as much. sub in your y value of 2 to solve f'(1).
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10/10/20
Michelle N.
When you're evaluating x=1, how did you get 2 in both (xy) and what happened to the y?09/27/20