Explained: "What is the instantaneous rate of change of..."

Jerimiah Boone

Answered question

2022-03-15

What is the instantaneous rate of change of

f (x) = (x^{2} - 2) e^{x^{2} - 3}

at x=2?

suboctavaja5

Beginner2022-03-16Added 6 answers

This is the same as evaluating f'(2)
Differentiate using the Product rule
If f(x) = g(x)*h(x) then f'(x) = g(x)*h'(x) + h(x)*g'(x)
here:

g (x) = (x^{2} - 2) \Rightarrow g^{'} (x) = 2 x

and

h (x) = e^{x^{2} - 3} \Rightarrow h^{'} (x) = e^{x^{2} - 3} \cdot \frac{d}{d x} (x^{2} - 3) = 2 x e^{x^{2} - 3}

now substitute these values into f'(x)

f^{'} (x) = (x^{2} - 2) \cdot 2 x \cdot e^{x^{2} - 3} + e^{x^{2} - 3} \cdot 2 x

factorising to obtain

f^{'} (x) = 2 x \cdot e^{x^{2} - 3} [x^{2} - 2 + 1]

\Rightarrow f^{'} (2) = 4 e^{1} (3) = 12 e \approx 32.62

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