İf they have same tangent line ,why their slopes are different .I don't understand.

Question

Lep $P=(x_1,y_1)$ be a point taken on the curve $y=x^3$.

The tangent line of this curve at $P$ meets the curve again at another point $Q=(x_2,y_2)$.

Prove that the slope of the curve at $Q$ is four times the slope at $P$.

How can I prove it ?

Answer 1

They don't have the same tangent line. You have a line $L$ which is tangent to the curve at $P$. It hits the curve in a second point, $Q$. But it isn't tangent to the curve at $Q$: the tangent at $Q$ is a different line $L'$ say.

Answer 2

They don't have the same tangent line. The tangent line is tangent at $P$(by definition), but at $Q$ the graph of the curve and the tangent at $P$ cross themselves.

For example look here, The point $P$ where it is tangent, whould be $(1,1)$ but in the other $Q$ it wouldn't.