Lesson 10 of 12
Interactive lesson · derivatives
Tangent Line at a Point: Derivative as Slope
At any point on a smooth curve, the tangent line shows the curve’s local direction. Move the point and the tangent changes with it. Its slope is the derivative at that input.
Learning goalInterpret f′(x) as the tangent slope at x.
f(x)=x2 → f′(x)=2x
Move the idea
Try it: move along the curve
Move the slider
Change one parameter and watch what changes with it.
The derivative is not just a rule. At each x, it is the slope you can see on the curve.
Local, not global
The curve can be steep in one place and flat in another. A derivative attaches a slope number to every input.
Sign tells direction
Positive slope means the function is rising locally; negative means falling; zero means the tangent is horizontal.
A new function
Collect all those slope numbers and you get another function, f′. For x², the moving readout traces the line 2x.
Step 3 · One-question check
Did it click?
For f(x)=x², what should happen to the tangent slope when x moves from 1 to 2?
Next: build the running-total idea before introducing integrals