Derivative formulas through geometry | Chapter 3, Essence of calculus

Calculating derivatives stops being a memorization exercise when each rule is tied to a single geometric idea: a derivative measures how a quantity changes under an infinitesimal nudge. The through-line is that tiny changes—dx and the corresponding tiny change in output—determine the slope-like ratio dF/dx. That perspective matters because most real-world models in calculus are built from familiar “pure” functions—polynomials, exponentials, trig functions—so learning how their derivatives behave gives a practical language for rates of change in concrete situations.

The discussion begins with f(x)=x². A tangent-line viewpoint suggests the slope increases with x, but the exact formula comes from unpacking what x² means geometrically. Interpret x as the side length of a square; then f(x) is the square’s area. Increasing x to x+dx adds a thin strip of area along two sides plus a tiny corner square. The dominant new area scales like 2x·dx, while the corner contributes dx², which becomes negligible compared with dx when forming the ratio dF/dx. That leaves d(x²)/dx = 2x, and the slope interpretation becomes precise: the derivative is the “area gained per unit length added.”

Next comes f(x)=x³, treated as the volume of a cube with side x. Nudging x to x+dx adds a yellow volume that, as dx shrinks, is overwhelmingly made of three thin “face” slabs. Each slab has area x² and thickness dx, so the leading volume change is 3x²·dx; edge and corner contributions scale like dx² or dx³ and vanish after dividing by dx. The result is d(x³)/dx = 3x², matching the idea that the graph’s slope at each point is exactly 3x².

From these cases, the power rule emerges: d(x^n)/dx = n·x^(n−1). The geometric reason is an expansion logic. When x is nudged to x+dx, the product (x+dx)^n has a first term x^n (the “old” volume/area), and the leading change comes from choosing exactly one dx factor and the remaining n−1 factors as x. There are n ways to pick which factor contributes dx, producing n·x^(n−1)·dx as the dominant term; all other terms contain dx² or higher and disappear in the dx→0 limit.

The same geometric mindset is applied to f(x)=1/x. Instead of treating 1/x as x^(−1) and hopping the exponent down, the function is pictured as a rectangle (a “puddle”) of fixed area 1: width x forces height 1/x. Increasing x by dx adds area on the right, so the height must drop by an amount d(1/x) that cancels the gained area. That cancellation yields a negative derivative consistent with the power-rule outcome. The segment also invites the reader to reason similarly for √x.

Finally, sine is handled through the unit circle. Walking an arc length θ on a radius-1 circle makes sin(θ) the vertical height. A tiny step dθ along the circumference changes the height by d(sin θ). Zoomed in, the circle locally resembles a straight line, creating similar right triangles whose geometry shows that d(sin θ)/dθ equals the adjacent-over-hypotenuse ratio—exactly cos(θ). The video closes by suggesting the same geometric approach can extend to derivatives of sums, products, and compositions in the next chapter.