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Differentiation Introduction
Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant. The slope of a curve at a point tells us the rate of change of the quantity at that point. The process of finding a derivative is called differentiation.
Rules for Differentiation
The Constant Rule
If y = c where c is a constant,
dy/dx = 0
e.g. y = 10 then
dy/dx = 0
dy/dx = 0
The Linear Function Rule
If y = a + bx
dy/dx = b
e.g. y = 10 + 6x then
dy/dx = 6
The Power Function Rule
If y = axn, a & n are constants
dy/dx = n . a . xn-1
i) y = 4x =>
dy/dx = 4 x 0 = 4
ii) y = 4x2 =>
dy/dx = 8 x
iii) y = 4x3 =>
dy/dx = 12 x2
iv) y = 4x-2 =>
dy/dx = - 8 x - 3
The Sum-Difference Rule
If y = f(x) ± g(x)
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If y is the sum/difference of two or more functions of x: differentiate the 2 (or more) terms separately, then add/subtract.
(i) y = 2x2 + 3x then
dy/dx = 4 x + 3
(ii) y = 4x2 - x3 - 4x then
dy/dx = 8 x - 3 x 2 - 4
(iii) y = 5x + 4 then
dy/dx = 5
The Product Rule
If y = u.v where u and v are functions of x Then
i) y = (x+2)(ax2+bx)
dy/dx = (x + 2)(2 ax + b)+(ax 2 + bx)
ii) y = (4x3-3x+2)(2x2+4x)
dy/dx = (4 x 3 - 3x +2)(2x2 + 4x)
dy/dx =(4x3 - 3 x + 2)(4 x + 4) + (2 x 2 + 4 x)(12 x2 - 3)
The Quotient Rule
If y = u/v where u and v are functions of x
The Chain Rule
If y is a function of v, and v is a function of x, then y is a function of x and
dy/dx = dy/dv . dv/dx
i) y = (ax2 + bx)
let v = (ax2 + bx), so y = v
dy/dx = 1/2 (ax 2 + bx )-1/2 . (2 ax + b)
ii) y = (4x3 + 3x 7 )4
let v = (4x3 + 3x 7 ), so y = v4
dy/dx = 4(4x3 + 3x - 7)3 . (12 x2 + 3)
The Inverse Function Rule
If x = f(y) then dy/dx = 1/ dx/dy
The derivative of the inverse of the function x = f(y), is the inverse of the derivative of the function
(i) x = 3y2 then
dy/dx = 6 y so dy/dx = 1/6y
(ii) y = 4x3 then
dy/dx = 12 x2 so dx/dy = 1/12 x2
Rule 9: Differentiating Natural Logs
if y = loge x = ln x =>
dy/dx = 1/x
NOTE: the derivative of a natural log function does not depend on the co-efficient of x
Thus, if y = ln mx => dy/dx = 1/x
Proof
if y = ln mx m>0
Rules of Logs => y = ln m+ ln x
Differentiating (Sum-Difference rule)
dy/dx= 0 + 1/x = 1/x
Example:
Find the equation of the normal to the curve y = 3 x - x at the point where x = 4.
Solution:
The curve can be written as y = 3x1/2 - x
Therefore, dy/dx = 3/2 x-1/2 - 1
When x = 4, y = 34 - 4 = 2
and dy/dx = 3/2*4-1/2 -1 = 3/2*1/2*-1 = -1/4
So the gradient of the normal is m = -1/ -1/4 = 4 .
To find the equation of the tangent:
y = mx + c => y = 4x + c
Substitute x = 4, y = 2: 2 = 4(4) + c i.e. c = -14.
So equation is y = 4x 14.
Important functions for differentiation:
Exponential and logarithmic functions:
d/dx ex = ex.
d/dx ax = In(a)ax.
d/dx In(x) = 1/x, x >0.
d/dx loga(x) = 1/x In(a).
Trigonometric functions:
d/dx sin(x) = cos(x).
d/dx cos(x) = -sin(x).
d/dx tan(x) = sec2 = 1/cos2(x) = 1 + tan2(x).
Inverse trigonometric functions:
d/dx arcsin(x) = 1/1-x2, -1 < x < 1.
d/dx arccos(x) = - 1/1-x2, -1 < x < 1.
d/dx arctan(x) = 1/1 + x2
