Find the derivative of f(x) =-10/x at x = -12
The lesson taught us to use the difference quotient but I can never get it to work with rational functions, can someone show me another way to do it or if I am doing it wrong?

Question

Find the derivative of f(x) =-10/x  at x = -12
The lesson taught us to use the difference quotient but I can never get it to work with rational functions, can someone show me another way to do it or if I am doing it wrong?

anonymous · Answer

@Cosmichaotic can u plz help?

anonymous · Answer

Sure. So, basically let's go ahead and graph the function f(x) = -10/x.

The derivative is the slope of the function at a given point (or rate of change). So we will look at the point x = -12 and take the limit as x approaches -12 from the left and from the right and see what slope we are approaching. This number, or slope, will be our derivative.

anonymous · Answer

Here is what the graph looks like in a professional sense.

anonymous · Answer

Let me expand the x values so we can see x = -12

anonymous · Answer

attachment

anonymous · Answer

Ohhh that makes a lot more sense

anonymous · Answer

So let's set x = -14 and x = -10 which is close to -12 on either side of it and calculate what our y-values are so that we can determine the slope.

Then we will do the same with x = -13 and x = -11

We will then see what the slope is approaching as it nears -12 from the positive and negative direction.

anonymous · Answer

Awesome. I'm sure there is an easier method of doing this, but this is the conceptual understanding of taking a limit to arrive at the derivative graphically.

anonymous · Answer

we plug them in the equation correct ?

anonymous · Answer

Correct. Plug in x = -14 into -10/x to get a point on the left side of -12
            Plug in x = -10 into -10/x to get a point on the right side of 12

With these 2 points you can calculate a slope using x2-x1/y2-y1 = m

The get two points closer to -12, which are x = -13 and -11 and do the same thing. Calculate the slope. What number is the slope approaching as we move closer and closer to -12 from the positive and negative direction?

anonymous · Answer

Correction on the top pat, I meant to say -12 not 12 when plugging in x = -10.

anonymous · Answer

Thank you so much I think I can get it now! :)

anonymous · Answer

Awesome @Venny. This isn't an easy concept for some and you got it on the first explanation - be proud of yourself for that!