The change of measure of function when the input of the function changes is called an ordinary derivative of a function. It is also defined as the insignificant change in the function with reference to one of its variables. An ordinary derivative for a specified input value defines the linear approximation of the function which is closer to that input value. At greater dimensions the ordinary derivative at a point is a linear transformation and is also called as the linearization.
Formulas of ordinary derivatives:
A simple ordinary derivative of a function f with respect to the variable x is indicated by df /dx.
The derivative of a function f which is given by the variable x is defined by,
f´(x) = lim f(x+h)-f(x) and it can also be calculated symmetrically in the form of
h->0 h
f´(x) = lim f(x+h)-f(x-h)
h->0 2 h
When a function (say f(x, y)) includes more than a single variable then the partial derivatives are used to specify the derivatives of those functions with respect to that variables.
Some of the ordinary derivatives of a function are mostly defined on the complex plane and they are referred to as the complex derivatives.
The second derivative of a function f with its variable x is given by,
f´´(x) = lim f(x+h)-f´ (x) and it can also be calculated symmetrically in the form of
h->0 h
f´´(x) = lim f(x+2h)-2f(x+h)+f(x)
h->0 h2
Simple derivatives of some functions:
`d/(dx)` (xn) = nxn-1
`d/(dx)` (ln x) = 1 / x
`d/(dx)` (sin x) = cos x
`d/(dx)` (cos x) = -sin x
`d/(dx)` (tan x) = sec2 x
Properties of Ordinary derivatives:
The derivatives of the sum are equal to the sum of the derivatives which is given by
[f(x) + …….+ h(x)]´ = f´(x) +…..+h´(x)
The product rule for the differentiation of the derivative is given by
`d/(dx)` [f(x) g(x)] = f(x) g´(x) + f´(x) g(x).
Here f´ is called as the ordinary derivative of the function f with respect to its variable x.
What are second derivatives:
Second derivatives involves the process differentiating the given algebraic function twice with respect to the given variable. Generally the derivative is discussed in calculus whereas it is mainly used to find the rate of change of the given function with respect to the change in the input. The following are the solved example problems with detailed step by step solution in second derivatives to study for the test.
Examples on second derivatives:
Ex 1: Find the second derivative for the function.
f(x) = 2x6 + 2 x5 + 3 x4 + 3x
Sol:
The given equation is
f(x) = 2x6 + 2 x5 + 3 x4 + 3x
The above function is differentiated with respect to x to find the first derivative
f '(x) = 2(6x 5) +2 (5 x4 ) +3(4 x3) + 3
By solving above terms
f '(x) = 12x 5 + 10x4 + 12 x3 – 3
The above function is again differentiated with respect to x to find second derivative
f ''(x) = 12(5x 4 ) – 10(4x3) + 12(3x2)
f ''(x) = 60x 4 – 40x3 +36x2 is the answer.
Ex 2: Find the second derivative for the function.
f(x) = 5x 2 +5x 4 + 12
Sol:
The given function is
f(x) = 5x 2 +5x 4 + 12
The above function is differentiated with respect to x to find the first derivative
f '(x) = 5(2x )+5(4 x 3 ) + 0
By solving above terms
f '(x) = 10x +20x3
The above function is again differentiated with respect to x to find second derivative
f ''(x) = 10(1 ) +20(3x2)
f ''(x) = 10 + 60x2 is the answer.
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Ex 3: Find the second derivative for the function.
f(x) = 4x4 +5x 5 +6x 6 + 2x
Sol:
The given function is
f(x) = 4x4 +5x 5 +6x 6 + 2x
The above function is differentiated with respect to x to find the first derivative
f '(x) = 4(4x 3 )+5(5x 4 ) +6( 6x 5) +2
By solving above terms
f '(x) = 16x 3 +25x 4 +36 x 5 + 2
The above function is again differentiated with respect to x to find second derivative
f ''(x)= 16(3x 2) +25(4x 3) +36 (5x 4)
f ''(x)= 48x 2 +100x 3 +180x 4 is the answer
Practice problems
1) Find the second derivative for the function.
f(x) = x 3 + x 4 + x 5
Ans : f ''(x) = 6x +12x2+ 20x 3
2) Find the second derivative for the function.
f(x) = 2x 3+3x5 + 4x 6
Ans : f ''(x) = 12x + 60x3 + 120 x 4
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