3 - Calculus

Question

Use limits to give mathematical definitions for each of the following derivatives, first with the $h\to 0$ definition of the derivative, and then with the $z\to x$ definition:

(a) the derivative of a function $f$ at the point $x=5$

(b) the derivative of a function $f$

(c) the right derivative of a function $f$ at the point $x=-2$

Step-by-Step Solution

Answer

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Step 1: h → 0 definition

The derivative of \(f\) at \(x = a\) using the \(h \to 0\) definition is:
\(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\)

Step 2: x → a definition

Equivalently, using the \(x \to a\) definition:
\(f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}\)