Question

What is the instantaneous rate of change of `f(x)=(x^2-2)e^(x^2-3)` at `x=2`?

Answer 1

12e ≈ 32.62

Explanation:

This is the same as evaluating f'(2)

Differentiate using the `color(blue)" 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 ) rArr g'(x) = 2x`

and h(x) `= e^(x^2-3) rArr h'(x) = e^(x^2-3). d/dx(x^2-3)=2xe^(x^2-3)`
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now substitute these values into f'(x)

f'(x) = `(x^2-2).2x.e^(x^2-3) + e^(x^2-3).2x`

factorising to obtain

f'(x)`= 2x.e^(x^2-3)[x^2-2+1]`

`rArr f'(2) = 4e^1(3) = 12e ≈ 32.62`