Using Derivatives to Solve Problems

Derivatives don't just live in textbooks—they help us solve real-world puzzles, from physics to economics.

Finding Maximum and Minimum Values

Derivatives help us locate the highest (maximum) or lowest (minimum) points of a function. These are super useful for optimizing things like profit, safety, or efficiency.

• Set the derivative to zero: \( f'(x) = 0 \).
• Use the second derivative to check if it's a max or min.

Related Rates

Some problems involve quantities changing together, like the length and area of a growing square. Derivatives help relate these rates of change.

Linear Approximations

Sometimes we use the tangent line (the derivative) to estimate a function's value near a point. It's like zooming in for a close-up!

Real-Life Impact

From finding the fastest route on a map to setting cruise control, derivatives make life smoother and more efficient.