90. Find f'(0) for
f(x) = e^(-1/x²), x≠0
= 0, x = 0.

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Recognize that the function is defined piecewise: \( f(x) = e^{-1/x^2} \) for \( x \neq 0 \) and \( f(0) = 0 \). We need to find the derivative at \( x = 0 \), i.e., \( f'(0) \).

Recall the definition of the derivative at a point: \( f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} \). Substitute the given function values to get \( f'(0) = \lim_{h \to 0} \frac{e^{-1/h^2} - 0}{h} = \lim_{h \to 0} \frac{e^{-1/h^2}}{h} \).

Analyze the behavior of the numerator \( e^{-1/h^2} \) as \( h \to 0 \). Since \( 1/h^2 \to \infty \), \( e^{-1/h^2} \to 0 \) very rapidly (faster than any polynomial rate).

Use this rapid decay to evaluate the limit \( \lim_{h \to 0} \frac{e^{-1/h^2}}{h} \). Consider whether the numerator approaches zero faster than the denominator approaches zero, which will determine the limit.

Conclude the value of \( f'(0) \) based on the limit evaluation, confirming whether the derivative exists and what its value is.

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Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change or the slope of the tangent line at that point. It is defined as the limit of the difference quotient as the increment approaches zero, i.e., f'(a) = lim(h→0) [f(a+h) - f(a)] / h.

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Functions like f(x) = e^(-1/x²) exhibit rapid decay to zero as x approaches zero from either side, due to the exponent tending to negative infinity. Understanding this behavior helps analyze limits and continuity near points where the function is defined piecewise.

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When a function is defined differently at a point (e.g., f(0) = 0) than elsewhere, the derivative at that point must be found using the limit definition, considering the function's behavior approaching that point. This often involves careful limit evaluation to determine differentiability and the derivative's value.

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