An introductory course that explores the fundamental concepts and techniques of mathematical analysis.
The derivative measures how a function changes as its input changes. It's like asking, "How fast is my position changing at this exact moment?"
The derivative of \( f \) at \( x \) is: \[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \] It tells us the instantaneous rate of change.
The derivative at a point gives the slope of the tangent line to the function's graph at that point.
Derivatives help us understand motion, optimize designs, and solve countless problems in science and engineering!
\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
If \( f(x) = x^2 \), then \( f'(x) = 2x \).
The derivative of \( f(x) = \sin x \) is \( \cos x \).
The derivative tells us how a function is changing at any instant — its rate of change.