How To Find Slope Of Vertical Line

The difference quotient should accept a greatcoat and boots because it has such a useful super-power: it gives you the slope of a curve at a unmarried signal. Want to know how?

Showtime, look at this effigy.

Graph of a parabola and a tangent line.

You lot can meet that the slope of the parabola at (7, nine) equals three, the gradient of the tangent line. Just yous can't calculate that slope with the algebra slope formula

because no matter what other point on the parabola you use with (vii, 0) to plug into the formula, yous'll go a slope that's steeper or less steep than the precise slope of 3 at (7, ix).

Now, what if your 2nd point on the parabola were extremely close to (seven, ix) — for example,

In this example, your line would exist almost exactly as steep equally the tangent line. The difference quotient gives the precise slope of the tangent line by sliding the second point closer and closer to (7, 9) until its distance from (seven, 9) is infinitely small.

Okay, plenty of this mumbo jumbo; now for the math. Here's the definition of the derivative based on the difference quotient:

Note that, as with well-nigh limit problems, plugging the arrow-number in at the beginning of a divergence quotient problem won't aid considering that gives you

As you'll see in the following practice questions, you lot accept to do a fiddling algebraic mojo so that you can cancel the h and and then plug in.

Practice questions

Use the deviation caliber to make up one's mind the derivative of the line y = four10 – 3.
Utilise the departure quotient to notice the derivative of the parabola f (x) = 310 ².