Implicit Differentiation
Some functions are of the type where separating dependent variable (y) and independent variable (x) is not possible these functions are of the form f(x,y) = 0 the differentiation of these functions is not found using the normal formulas then the differentiation of these functions is found using the concept as shown in the example added below,
Example: Find the differentiation of x2 + y2 + 4xy = 0
Solution:
x2 + y2 + 4xy = 0
Differentiating with respect to x,
2x + 2ydy/dx + 4(xdy/dx + y) = 0
⇒ 2x + 4y + 2dy/dx(y + 2x) = 0
⇒ x + 2y + dy/dx(y + 2x) = 0
⇒ dy/dx(y + 2x) = -(x + 2y)
⇒ dy/dx = -(x + 2y)/(y + 2x)
Higher Order Differentiation
Higher order differentiation is nothing, but the differentiation of a function more than one time suppose we have a function y = f(x) then its differential in higher order is calculated as,
First Derivative = dy/dx = f'(x)
Second Derivative = d2y/dx2 = f”(x)
Third Derivative = d3y/dx3 = f”'(x)
….
….
nth Derivative = dny/dxn = f(n)(x)
This can be understood using the example added below,
Example: Find the second-order derivative of f(x) = 4x4 + 3x3 + 2x2 + x + 1
Solution:
f(x) = 4x4 + 3x3 + 2x2 + x + 1
Differentiating with respect to x,
f‘(x) = 4(4x3) + 3(3x2) + 2(2x) + 1 + 0
⇒ f'(x) = 16x3 + 9x2 + 4x + 1
For second-order derivative differentiating with respect to x,
f”(x) = 16(3x2) + 9(2x) + 4 + 0
⇒ f”(x) = 48x2 + 18x + 4
This is the required second-order derivative.
Articles Related to Differentiation Formulas:
Differentiation Formulas
Differentiation Formulas: Differentiation allows us to analyze how a function changes over its domain. We define the process of finding the derivatives as differentiation. The derivative of any function f(x) is represented as d/dx.f(x)
In this article, we will learn about various differentiation formulas for Trigonometric Functions, Inverse Trigonometric Functions, Logarithmic Functions, etc., and their examples in detail.
Table of Content
- What is Differentiation?
- Differentiation Formula
- Basic Differentiation Formulas
- Differentiation of Trigonometric Functions
- Differentiation of Inverse Trigonometric Functions
- Differentiation of Hyperbolic Functions
- Differentiation Rules
- Differentiation of Special Functions
- Implicit Differentiation
- Higher Order Differentiation
- Examples of Differentiation Formulas
- Practice Problems on Differentiation Formulas