Question

Find the gradient of the secant joining the points P(a, f(a)) and Q(a+h, f(a+h)) ... Calculus!

Answer 1

Here is the problem

Answer 2

The gradient of the secant?

Answer 3

Yes :)

Answer 4

What does that mean?
I know what the secant line is, but not what the gradient of it is...

Answer 5

Would it be the slope?

Answer 6

Gradient is slope, yes

Answer 7

Here is the problem, if its easier for you to see.
I know the answer, but dont know how to get it...

Answer 8

Well, basically you want to find: \[
\frac{\Delta y}{\Delta x} = \frac{Q_y-P_y}{Q_x-P_x} = \frac{f(x+h)-f(x)}{(x+h)-(x)} = \frac{f(x+h)-f(x)}{h}
\]

Answer 9

Yepp, I did that, but my answer was not what the textbook suggested..

Answer 10

Can you show me your work a bit?

Draw it or latex it.

Answer 11

\[f(x+h) = 1/\left\{ (x+h)-1 \right\} \]
So
\[f'(x) = \left[ 1/\left\{ (x+h)-1 \right\} - 1/(x-1)\right] / h\]

Sorry is that too hard to read?

Answer 12

No that's fine.

Answer 13

Now let's focus on this: \[
\frac{1}{x+h-1}-\frac{1}{x-1}
\]How can we simplify it?

Answer 14

Oh, do you cross multiply the denominator?
I think I got it, thank you!