It is given that $y = f(g(x))$, where $f(1) = 3$, $f'(1) = -4$, $g(5) = 1$ and $g'(5) = 2$ - HSC - SSCE Mathematics Advanced - Question 7 - 2023 - Paper 1
Question 7
It is given that $y = f(g(x))$, where $f(1) = 3$, $f'(1) = -4$, $g(5) = 1$ and $g'(5) = 2$.
What is the value of $y'$ at $x = 5$?
Worked Solution & Example Answer:It is given that $y = f(g(x))$, where $f(1) = 3$, $f'(1) = -4$, $g(5) = 1$ and $g'(5) = 2$ - HSC - SSCE Mathematics Advanced - Question 7 - 2023 - Paper 1
Step 1
Find $g(5)$
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Answer
From the given information, we know that g(5)=1.
Step 2
Find $f'(g(5))$
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Answer
Since g(5)=1, we need to find f′(1). We are given that f′(1)=−4.
Step 3
Find $g'(5)$
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Answer
We are directly provided that g′(5)=2.
Step 4
Use the chain rule to find $y'$
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Answer
Using the chain rule, we know that:
y′=f′(g(x))imesg′(x).
At x=5, this becomes:
y′=f′(g(5))imesg′(5)=f′(1)imesg′(5)=−4imes2=−8.
Step 5
Final Answer
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