Calculus 1 introduces the fundamental concepts of limits, derivatives, and integrals, providing a foundation for advanced mathematical studies.
The derivative measures how a function changes as its input changes. Think of it as the "speed" or "slope" of a curve at any point. If you're driving a car, the speedometer shows the derivative of your position with respect to time!
The derivative of \( f(x) \) is written as \( f'(x) \) or \( \frac{df}{dx} \).
The formal definition uses limits:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
The derivative at a point gives the slope of the tangent line to the function at that point.
The derivative of \( x^2 \) is \( 2x \).
The derivative of \( \sin(x) \) is \( \cos(x) \).
Derivatives tell us how things change from one moment to the next.