The function g(x)=ln(f(x)) passes through the origin. Show why the derivatives of f(x) and g(x) are equal when x = 0. Show all your work.
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Step-by-step explanation:
g(x) = ln(f(x))
Take derivatives of both sides.
g'(x) = 1/f(x) f'(x)
g(x) passes through the origin, so g(0) = 0:
g(0) = 0
ln(f(0)) = 0
f(0) = 1
Therefore:
g'(0) = 1/f(0) f'(0)
g'(0) = 1/1 f'(0)
g'(0) = f'(0)
Answer:
Since at x = 0, g(x) = 0
ln(f(x)) = 0
f(x) = e^0 = 1
At x = 0, f(x) = 1
g'(x) = (1/f(x)) × f'(x)
At x = 0,
g'(x) = (1/1) × f'(x)
g'(x) = f'(x)