MEI Core 2 Differentiation Chapter Assessment

I am finding this question rather tricky, would anyone be able to help me?

10. The points P(x,y) and Q(x+dx,y+dy) lie on the curve y=3x^2 +2

a) Show that the gradient of the chord PQ is (3(x+dx)^2 - 3x^2)/ dx.

b) Simplify this expression.

c) Hence prove from first principles that the derivative of 3x^2 is 6x.

*note: d's above are meant to be delta symbols.

Part a) I have done, and for part b) I worked out as being 6+3dx (not sure it is correct!). I have no idea on how to do c).

Thanks for your time.

Reply 1

Fermat

Delta Seven

I am finding this question rather tricky, would anyone be able to help me?

*note: d's above are meant to be delta symbols.

Part a) I have done, and for part b) I worked out as being 6+3dx (not sure it is correct!). I have no idea on how to do c).

Thanks for your time.

(a)
gradient of PQ is m, where
m = {(y+Δy) - y}/{(x+Δx) - x}
y=3x^2 +2, so
y+Δy = 3(x+Δx)² + 2
y = 3x² + 2, so
(y+Δy) - y = 3(x+Δx)² - 3x²
=====================
and
(x+Δx) - x = Δx
============

substituting for (y+Δy) - y and (x+Δx) - x in the expressoiun for m,
m = {3(x+Δx)² - 3x²}/Δx
===================
(b)
expanding the expressoin for m,

m = {3(x² + 2xΔx + (Δx)²) - 3x²} / Δx
m = {6xΔx + 3(Δx)²} / Δx
m = 6x + 3Δx
==========
(c)
From first principles, The derivative of the slope, at the point P, is given by the slope of the chord PQ, where the point Q moves along the curve towards the point P. Then the slope, m, tends towards the slope of the curve at the point P - which is the derivative of the curve at the point P.

as Q tends towards P,
Δx tends towards zero
m tends towards 6x
Hence the derivative of the curve is 6x
=============================

Reply 2