Differentiation. Newton. An Introduction. Leibniz. . Q. Find the gradient of the curve f(x)=x 2 at the point P (x, f(x)). f(x +h). . P. Draw a tangent at P and try and find the gradient of this tangent. f(x). x. x + h. Approximate this tangent by a chord PQ where Q=(x+h,f(x+h)).
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Find the gradient of the curve f(x)=x2
at the point P(x, f(x))
Draw a tangent at P and try and find the gradient of this tangent.
x + h
Approximate this tangent by a chord PQ where Q=(x+h,f(x+h))
Now we can find the gradient of this chord ….
But as h approaches 0 we can see that 2x + h approaches 2x
We say that 2x is the limit of 2x+ h as h approaches 0 and write this as:
We call this gradient function the derived function or derivative
We denote the derivative of f(x) by f’(x)
The process of finding the derivative is called differentiation
This means that if at the point x=3 the slope of the curve is 2(3)=6
f(x) = x4
f(x) = 4x3
If f(x)=axn +bxm
Then f’(x)=naxn-1 + mbxm-1
f(x)=3x-2 - x-3
f’(x)= -6x-3 +3x-4
The derivative is sometime written using a notation introduced by Leibniz
y = x4
The derivative measures the slope of a curve. When the curve represents a relation between variable x and a dependant variable y we interpret the derivative as the rate of change of y with respect to x
Gradient =Rate of Change of y with respect to x
We know the equation of a line is :
If P(a , b) is a point on the line then we can find C
.P(a , b)
Therefore the equation of the gradient at P
The normal is the line perpendicular to the tangent
Gradient of tangent x gradient of normal =-1
.P(a , b)
The second derivative measures the slope of the first derivative. We write it like this: