Free maths tool · Level: Ages 16 to 18
Derivative and tangent line
Drag the point along the curve and watch the tangent tilt with it. The derivative is just the slope of that line.
The derivative of a function at a point is the slope of the tangent line there. You find it by taking the slope of the secant between two points and sliding those points together. The smaller h gets, the closer the secant sits to the tangent.
Why h can never be zero
The slope of a secant is (f(x+h) - f(x)) / h. Set h to zero and you divide by zero, which is not allowed. But nothing stops you making h as small as you like.
Slide h down and follow the two numbers on the right. The secant and the tangent creep towards each other without ever meeting. That creeping is exactly what a limit is, which is why the definition says lim h→0.
The derivative is a function too
Every point on the curve has its own slope. Collect all of those slopes and you have a new function: the derivative.
That is what the second graph shows. Drag the point left and right and watch where the derivative crosses zero. That is where the original function has a peak or a trough, because the tangent is horizontal there.
What the signs tell you
Positive derivative, the function is rising. Negative derivative, it is falling. That is the whole basis of a sign table, and you can read it off the direction the tangent points.
Try f(x) = x³ - 3x. Between the two turning points the tangent clearly slopes down and the derivative sits below the axis. You do not need to memorise that link once you have watched it move.
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Frequently asked questions
- What is a derivative in plain language?
- How steep a graph is at a particular spot. Rest a ruler against the curve so it just touches: the slope of that ruler is the derivative there.
- What is the difference between a secant and a tangent?
- A secant cuts the curve at two points, a tangent touches it at one. Slide the two crossing points of the secant together and in the limit you get the tangent.
- How do I find the equation of the tangent line?
- You need a point and a slope. The point is (a, f(a)) and the slope is f’(a). Put them into y = f’(a)(x - a) + f(a) and you are done.
- Why is the derivative zero at a maximum?
- At a maximum the function switches from rising to falling. At that switch the tangent lies flat, and a flat line has slope zero.