Differentiation
Introduction to Differentiation
Differentiation is a method used to compute the rate of change of a function f(x) with respect to its input x. It is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input.
For an equation plotted on a graph, the steepness of that graph's slope is called its derivative.
Let a function f(x) take in an input of x. The rate of change of f(x) per unit change in x is the derivative of that function. There are several ways to provide notation: f'(x), dy/dx, y' etc.
Linear and Non-linear Functions
Functions are classified as two types in calculus:
- Linear
- Non-linear
A linear function varies at a constant rate through its domain. The overall rate of change of the function is the same as the rate of change of a function at any point. It varies in the case of non-linear functions.
Brief Understanding of Limits
In mathematics, a limit is the value that a function approaches as its argument approaches some value. Limits of functions are essential to calculus. They define continuity, derivatives and integrals.
If a function undergoes an infinitesimal change of 'h' near any point 'x', then the derivative of the function is defined as:
Rules for Derivatives
Sum/Difference rule for derivates:
If f(x) = u(x) + v(x) => f'(x) = u'(x) + v'(x)
Chain rule:
If y = f(x) = g(u) and u = h(x)
dy/dx = dy/du × du/dx
Common Derivatives:
Using these formulae, the derivatives of functions can be calculated If a function is differentiated once, the derivative is a first order derivative f ′(x) If the first order derivative is further differentiated, the function obtained is second order derivative f ′′(x)
Finding Maximum and Minimum Values
Using derivatives, the maximum and minimum value of a given function can also be calculated. For a function f(x), at one point, x will give the maximum and minimum value of the function f(x).
- Equate the first order derivative to 0 i.e. f'(x) = 0 and obtain the value of x=c
- Then substitute this value in the second order derivative by further differentiating
- If f''(x) < 0, the maximum value of f(x) is when x=c
- If f''(x) > 0, the minimum value of f(x) is when x=c
Examples
Example 1:
Calculate the maximum possible area of a rectangle that has a perimeter of 116 meters?
Let the length and width be L and W
P = 2L + 2W => 116 = 2L + 2W => = 58 − L
A = L * W => A = L (58 − L) => A = 58L − L2
Let f (x) = 58L − L2 => f ′(x) = 58 − 2L
But f’(x) = 0 => 58 − 2L = 0 => L = 29
Now to find the maximum Area the second derivative is obtained:
f ′(x) = 58 − 2L => f ′′(x) = − 2
As −2 < 0, f’(x) is maximum when L=29
L=29,=58 − L=> = 29
Area A = WL = 29(29) = 841 => Maximum possible Area is 841 m2
Example 2:
Differentiate f(x) = 10x² with respect to x
Using Power rule i.e. Bullet 1 from fig. 1
f ′(x) = 10(2)x1 = 20x => Derivative of f (x) = 10x2 is f ′(x) = 20x
Example 3:
Differentiate f(x) = sin(3x + 5) with respect to x
From fig.1 for f(x) = sin x, f’(x) = cos x
In this equation we have an inner (sin) and an outer (3x + 5) function
Hence we must first differentiate the inner function first and then multiply that with the derivative of the outer function
=> f ′(x) = cos(3x + 5) * (3) => f ′(x) = 3cos(3x + 5)
Example 4:
Differentiate f(x) = tan²x with respect to x
From fig.1 first power rule must be applied to the trigonometric function, and then the trigonometric function should be differentiated
=> f ′(x) = 2tan1x * sec2 x (if f(x) = tan x, f ′(x) = sec2 x) => f ′(x) = 2tan xsec2 x
Example 5:
Determine the slope of the curve f(x) = x³ - x² + 1 at the given point (2, -15)
As mentioned, the slope of a curve is the first order derivative of the equation of the curve To find the slope of this curve at the point (2, -15):
f ′(x) = 3x2 − 2x + 0 => f ′(x) = 3x2 − 2x
At the point (2, -15), substitute the values in the slope equation
3(2)2 − 2(2) = 12 - 4 = 8 => Slope of the curve at the (2, -15) is 8
Excersise Questions
