The Derivative Machine

Calculus 1 · Axiom Academy

Functions go in, slope functions come out. Differentiation is a machine you can learn to operate. One machine sits behind all of calculus Picture a machine with one job: you feed in a function f(x) , and out the other side comes a brand-new function — its derivative f'(x) , the function that reports f 's slope at every point. Master that one machine and a huge slice of calculus becomes turning a crank. Watch the simplest run: the input f(x)=x^2 drops into the machine, and out comes the output f'(x)=2x . The machine doesn't return a number — it returns a whole new curve , the slope function. In goes a function, out comes its slope function. That transformation — f f' — is what "taking a derivative" means. Feed the machine — read off the output The machine has buttons, one per kind of input. Press one to feed that function in; the machine applies its rules and hands back the derivative f'(x) — shown both as the symbolic output and as the output curve beside the input. Same machine every time — only the input changes. The rule it applies (power, sum, the constant rule) depends on what you feed it. The output value is the input's slope, point by point Why is 2x the "derivative" of x^2 ? Drag the point along the input curve f(x)=x^2 . At each spot the machine's output f'(x)=2x is exactly the slope of the tangent line resting on the curve there — read both and watch them stay equal.

This is the written version of the interactive lesson above. See the full Calculus 1 course.