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ussjt
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f '(8)=5 g '(8)=3 f(4)=8 g(4)=10 g(4)=10 g(8)=2 f(8)=5
find (g o f)'(4)
how do I go about setting up these types of problem.
find (g o f)'(4)
how do I go about setting up these types of problem.
ussjt said:what about this type of problem:
For a given functionhttps://webwork2.math.ohio-state.edu/courses/math151wi06sm/tmp/png/Hmwk5/696680/kopko.4-prob5image1.png consider[/URL] the composite function https://webwork2.math.ohio-state.edu/courses/math151wi06sm/tmp/png/Hmwk5/696680/kopko.4-prob5image2.png Suppose we know that https://webwork2.math.ohio-state.edu/courses/math151wi06sm/tmp/png/Hmwk5/696680/kopko.4-prob5image3.png
Calculate f ' (x)
How do I go about setting up this type of problem?
It's fine up to here.ussjt said:Here are my steps:
f '(2x^3)(6x^2)= 7x^5
f '(2x^3)= (7x^5)/(6x^2)
f '(2x^3)= (7x^3)/6
~~~~~~~~~~~~
2x^3=y
x^3= y/2
where did the f'(y) come from?VietDao29 said:It's fine up to here.
Now sub what you get in the expression:
f '(2x3)= (7x3)/6, we have:
f'(y) = 7y / 12.
So what's f'(x)?
Can you go from here?
ussjt said:where did the f'(y) come from?
The chain rule is a mathematical rule that allows us to find the derivative of a composite function, which is a function that is made up of two or more functions. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.
To apply the chain rule, we first need to find the derivatives of the individual functions, f and g. Then, we can plug in the given values for f'(8) and g'(8) and substitute f(4) and g(4) into the composite function (g o f)(x). Finally, we can use the chain rule formula to find the derivative of the composite function at x=4.
(g o f)'(4) represents the derivative of the composite function (g o f) at x=4. Essentially, it tells us the rate of change of the composite function at that point, which is the result of combining the two individual functions, f and g.
To find the derivative of a composite function, we first find the derivatives of the individual functions involved. Then, we plug in the given values for those derivatives and the inputs of the composite function. Finally, we use the chain rule formula to calculate the derivative of the composite function at the given input.
The given information, f'(8)=5, g'(8)=3, f(4)=8, and g(4)=10, provides us with all the necessary information to apply the chain rule formula and find the derivative of the composite function at x=4. By plugging in these values and using the chain rule, we can solve for (g o f)'(4) and determine the rate of change of the composite function at that point.