Techniques of Differentiation
Some other techniques of differentiations are:
- Implicit Differentiation
- Logarithmic Differentiation
- Parametric Differentiation
Implicit Differentiation
Implicit differentiation is used when you have an equation involving both [Tex]x[/Tex] and [Tex]y[/Tex] that does not explicitly solve for y. In such case you differentiate both sides of the equation with respect to ( x ) and then solve for [Tex]\frac{dy}{dx}[/Tex]
For example:
Given [Tex]x^2 + y^2 = 1[/Tex] differentiate both sides with respect to [Tex]x[/Tex]:
[Tex]\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(1)[/Tex]
Applying the chain rule to [Tex]y^2[/Tex] , we get:
[Tex]2x + 2y\frac{dy}{dx} = 0[/Tex]
Solving for [Tex] \frac{dy}{dx}[/Tex] we find:
[Tex]\frac{dy}{dx} = -\frac{x}{y}[/Tex] .
Logarithmic Differentiation
Logarithmic differentiation is useful when dealing with products, quotients or powers of functions that would be cumbersome to differentiate using standard rules. You take the natural logarithm of both sides of an equation and then differentiate.
Example: Consider [Tex]y=x^x[/Tex].
Solution:
To differentiate this function take the natural log of both sides:
[Tex]\ln(y) = \ln(x^x)[/Tex]
Apply the properties of logarithms:
[Tex]\ln(y) = x\ln(x)[/Tex]
Differentiate implicitly with respect to [Tex]x[/Tex]
[Tex]\frac{1}{y}\frac{dy}{dx} = \ln(x) + 1[/Tex]
Solve for ( [Tex] \frac{dy}{dx}[/Tex]):
[Tex]\frac{dy}{dx} = y(\ln(x) + 1) = x^x(\ln(x) + 1)[/Tex]
Parametric Differentiation
When a curve is defined parametrically by two equations [Tex]x=x(t)[/Tex] and [Tex]y=g(t)[/Tex] , one can use parametric differentiation to find derivative [Tex]\frac{dy}{dx}[/Tex].
The derivative [Tex]\frac{dy}{dx}[/Tex] can be calculated by dividing [Tex]\frac{dy}{dt}[/Tex] by [Tex]\frac{dx}{dt}[/Tex] ;
[Tex]\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}[/Tex]
For example Let [Tex] x = \cos(t) [/Tex] and [Tex]y = \sin(t)[/Tex], then
[Tex]\frac{dx}{dt} = -\sin(t), \quad \frac{dy}{dt} = \cos(t)[/Tex]
So, the derivative [Tex]\frac{dy}{dx}[/Tex] is
[Tex]\frac{dy}{dx} = \frac{\cos(t)}{-\sin(t)} = -\cot(t)[/Tex]
Fundamental of Differential Calculus
Differential calculus is a branch of calculus that studies the concept of a derivative and its applications. Derivative tells us about the rate at which a function changes at any given point. Differential Calculus is crucial to many scientific and engineering areas since it allows for the estimation of instantaneous rates of change and curve slopes. In this article, we will be discussing about Differential Calculus and its fundamentals, which every students should know.
This article helps learners in understanding of differential calculus, its concepts, and its applications. By the end of this article, readers should be able to understand the fundamentals of derivatives and use them to solve real-world issues.
Table of Content
- Key Concepts in Differential Calculus
- Limits
- Continuity
- Derivatives:
- Differentiation Notation
- Basic Rules of Differentiation
- Product Rule of Derivative
- Quotient Rule of Derivatives
- Sum Rule of Derivative
- Power Rule of Derivative
- Constant Multiple Rule of Derivative
- Chain Rule of Derivative
- Differentiation of Common Functions
- First Principle of Differentiation
- Techniques of Differentiation
- Applications of Differential Calculus