Basic Rules of Differentiation

To find the derivative of more complicated functions, we have some rules that make the derivative more simple and easy. Some of them are:

• Product Rule
• Quotient Rules
• Sum Rule
• Power Rule
• Constant Multiple Rule
• Chain Rule

Let’s discuss these rules in detail as follows:

Product Rule of Derivative

Product rule is a formula for calculating the derivative of a function product of two or more functions. The derivative of a product of two functions is defined as the first function multiplied by the derivative of the second function + the second function multiplied by the derivative of the first function.

If we have two functions, u(x) and v(x), and y = u(x)v(x), then the derivative of y with respect to x is given by:

[Tex]\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx} [/Tex]

This rule applies when working with a function that is a combination of two or more simpler functions. It is a necessary technique in calculus for differentiating complicated functions.

Quotient Rule of Derivatives

Quotient Rule is a calculus technique for determining the derivative of a function whose is the ratio of two differentiable functions. If given function f(x) is Quotient of two functions which are differentiable with respect to x i.e.,

[Tex]f(x)=\frac{h(x)}{g(x)}[/Tex]

Its derivative, written as f'(x), is given by:

[Tex]f'(x) = \frac{g(x).h'(x) – h(x). g'(x)}{{g(x)}^2}[/Tex]

This indicates that the derivative of a function ratio (the first function divided by the second function) is equal to the ratio of (the derivative of the first function times the second function minus the derivative of the second function times the first function) to the square of the second function.

Sum Rule of Derivative

The sum rule of derivative is a formula for calculating the derivative of a function sum of two or more functions. If we have two functions, u(x) and v(x), and y = u(x)+v(x), then the derivative of y with respect to x is given by:

y'(x)=u'(x)+v'(x)

Power Rule of Derivative

Power Rule is a calculus technique for determining the derivative of a function whose have algebraic expression of power n. If given function f(x) = xⁿ, where n is real number, then derivative of the function is given by:

[Tex]f^{\prime}(x)=nx^{(n-1)}[/Tex]

Constant Multiple Rule of Derivative

If we have functions, y (x) = c f(x) where c be a constant, then the derivative of y with respect to x is given by:

[Tex]y^{\prime}(x)=c\times f^{\prime}(x)[/Tex]

Chain Rule of Derivative

Chain rule is a formula for calculating the derivative of a composite function. If we have composite functions, y(x)=f(g(x)) where f and g are function of x and differentiable. then the derivative of y with respect to x is given by:

[Tex]y^{\prime}(x)=f^{\prime}(g(x)).g^{\prime}(x)[/Tex]

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.