All About Derivatives

From Newton’s Vision to Modern Visualization

Back in the 1600s, Isaac Newton didn’t have computers or graphing software, yet he could picture something extraordinary. He imagined a tangent line moving along a curve, changing its slope point by point. We can now see exactly what Newton could only visualize in his mind, thanks to modern tools like Desmos. When we animate a tangent sliding along a graph, we notice how its gradient (slope) changes continuously. Newton realized that if there were a shortcut method to find that slope without drawing the graph, it would revolutionize how we study motion and change. That idea became one of the greatest breakthroughs in mathematics — the birth of CALCULUS.

Watch the tangent move “just like Newton imagined” Touch the Point of Tangency And Move It As You Wish

So our \( \displaystyle \frac{dy}{dx} \) is a gradient-generating function — it gives the slope of the tangent at any point and lets us visualize a curve, even before we draw it.

Hence Sign of \( \displaystyle \frac{dy}{dx} \) Tells Us

\( \displaystyle \frac{dy}{dx} \) is +ve

Curve is increasing
(rising as x increases)

\( \displaystyle \frac{dy}{dx} \) is –ve

Curve is decreasing
(falling as x increases)

\( \displaystyle \frac{dy}{dx} \) = 0

Curve has a turning point
(local max or min)

Now Scroll Up!!! and look at the animation
You will see that the tangent’s slope tells us whether the curve rises, falls, or reaches a maximum or minimum.