SFU Beedie courses only. See the course list →
Business TutorBusiness Tutor
← Everything for Calculus I
Calculus IDerivatives

The derivative, a slope at one instant.

Move a secant towards a tangent, turn slopes into a new function, and make sense of rates of change. Then solve the questions from your notes, one step at a time.

Slopes and rates · Step 1 of 9

Bring two points closer together

Burgundy: curve · Gold: secant · Dashed blue: tangent

−10123−202468PQ
A secant goes through two points on a curve. Its slope is the change in height divided by the change in horizontal position. A tangent describes the slope at just one point: it is what the secant approaches as the gap shrinks.
Try it: Drag the gap h{h} towards zero, then approach from the other side. The gold secant approaches the dashed blue tangent to f(x)=x2{f(x)} = {x^2} at (1,1){(1,1)}.
Size of the gap h1
Approach from
msecant=(1+h)2−1h=2+h=3.00{\displaystyle m_{\rm secant}} = {\displaystyle \frac{(1+h)^2-1}{h}} = {\displaystyle 2} + {\displaystyle h} = {\displaystyle 3.00}
lim⁡h→0(2+h)=2{\displaystyle \lim\limits_{h \to 0}(2+h)} = {\displaystyle 2}

We let the gap approach zero. We never divide by a gap of zero.

Check yourself
What does a secant slope measure?

Need a hand with tangent lines or the limit definition?

Bring your practice questions. We work through them together until every type feels routine.

Book a sessionNext topic
FAQ

Questions, answered.

Still unsure about something? Message us and you will hear back within a day.

See all questions

  • In person, we meet at the SFU Burnaby campus or in a private meeting room near Brentwood in Burnaby. Online sessions run on Zoom. You choose when you book.