Differentiation of quotient of two functions

Introduction

Two functions coupled by some operators can be easily discriminated based on the type of operator utilized in differentiation. If a positive sign is used between two functions, the functions can be separated individually using the +ve sign; the same can be said for a negative sign. The same cannot be said for a multiplication or division sign. In multiplication, each function is differentiated separately while the other function is treated as a constant, and the rule becomes a little more complicated if the functions are dividing. Let’s take a look at this differentiation rule, often known as the quotient rule.

Functions

A function is a method or a relationship that connects each member ‘a’ of a non-empty set A to at least one element ‘b’ of another non-empty set B. In arithmetic, a function is a relation from one set A (the domain of the function) to another set B (the co-domain of the function).

 f = {(a,b)| for all a ∈ A, b ∈ B}

Linear and Non-Linear Functions

A function that is not linear is easily defined as a non-linear function. As a result, knowing how to grasp a linear function is required before learning how to understand a non-linear function. The graph of a linear function is a line. A linear function in algebra is a polynomial with the highest exponent of 1 or a horizontal line (y = c, where c is a constant). A linear function is a function that has a continuous slope (rate of y with respect to x). This indicates that the slope of the line connecting any two points on the function is the same.

Now that we know what a linear function is, let’s define a non-linear function. As previously stated, non-linear functions are functions that are not linear. They have the polar opposite features of a linear function as a result. The graph of a linear function is a line. As a result, a non-linear function’s graph is not a straight line. Linear functions have a constant slope, whereas non-linear functions have a slope that varies across points. Linear functions are polynomials with the highest exponent of 1 or of the form y = c in algebra, where c is constant. The rest of the functions are non-linear.

Differentiation

Differentiation is the algebraic method for calculating derivatives. The derivative of a function is the slope or gradient of a certain graph at any given place. The gradient of a curve is the value of the tangent traced to it at every given point. For non-linear curves, the gradient varies at different points along the axis. As a result, determining the gradient in these circumstances is difficult.

It’s also known as a property’s change about the unit change of another property.

Consider the function f(x) as a function of the independent variable x. The independent variable x is caused by a tiny change in the independent variable Δx. The function f(x) undergoes a similar modification Δf(x). Ratio is:

Δf(x)Δx is a measure of f(x) of change in relation to x.

As Δx approaches zero, the ratio’s limit value is

limΔx0f(x)Δx  and is known as the first derivative of the function f(x).

Quotient Rule

The quotient rule is a calculus method for obtaining the derivatives of any function given as a quotient obtained by dividing two differentiable functions. According to the quotient rule, the ratio of the outcome is similar to the derivative of a quotient formed by subtracting the numerator multiplied by the denominator’s derivatives from the denominator multiplied by the denominator’s derivatives to the denominator’s square.

ddxf(x)g(x)=f'(x)g(x)-f(x)g'(x)[g(x)]2

Derivation of Quotient Rule using Derivative and Limit properties

Let’s assume, z(x)=f(x)g(x)

z'(x)=h0[z(x+h)-z(x)]h

h0f(x+h)g(x+h)-f(x)g(x)h

h0f(x+h)g(x)-g(x+h)f(x)h(g(x+h)g(x))

h0f(x+h)g(x)-g(x+h)f(x)hh01g(x+h)g(x)

h0f(x+h)g(x)-f(x)g(x)+f(x)g(x)-g(x+h)f(x)h1(g(x))2

h0f(x+h)g(x)-f(x)g(x)h-h0g(x+h)f(x)-f(x)g(x)h1(g(x))2

g(x)h0f(x+h)-f(x)h-f(x)h0g(x+h)-g(x)h1(g(x))2

f'(x)g(x)-g'(x)f(x)(g(x))2

Derivation of Quotient Rule using Implicit Differentiation

Let’s assume, z(x)=f(x)g(x)

f(x)=z(x)g(x)

f'(x)=z'(x)g(x)+z(x)g'(x)

Putting value of z(x)

f'(x)=z'(x)g(x)+f(x)g(x)g'(x)

f'(x)g(x)=z'(x)(g(x))2+f(x)g'(x)

f'(x)g(x)-f(x)g'(x)=z'(x)(g(x))2

z'(x)=f'(x)g(x)-f(x)g'(x)(g(x))2

Derivation of Quotient Rule using Chain Rule

Let’s assume, z(x)=f(x)g(x)

z(x)=f(x)g-1(x)

z'(x)=f'(x)g-1(x)+f(x)ddxg-1(x)

z'(x)=f'(x)g-1(x)+f(x)(-1)(g-2(x))g'(x)

z'(x)=f'(x)g(x)-f(x)g'(x)g(x)2

z'(x)=f'(x)g(x)-f(x)g'(x)g(x)2=ddxf(x)g(x)

How to apply quotient rule in differentiation

Both f(x) and g(x) must be differentiable functions in order to compute the derivative of the function z(x)=f(x)g(x). Using the quotient rule, we can determine the derivation of a differentiable function z(x)=f(x)g(x) by following the steps below.

Note down the values of f(x) and g(x).

Calculate the values of f'(x) and g'(x) and put the values in the quotient rule formula.

z'(x)=f'(x)g(x)-f(x)g'(x)g(x)2

Examples of Quotient Rule

1. z(x)=x5-cosxsinx

after using quotient rule,

z'(x)=x5-cosx’sinx-x5-cosxsinx’sinx2

z'(x)=5×4-sinxsinx-x5-cosxcosxsinx2

z'(x)=1+5x4sinx-x5cosxsinx2

  2. z(x)=x+cosxtanx

after using quotient rule,

z'(x)=x+cosx’tanx-x+cosxtanx’tanx2

z'(x)=1-sinxtanx-x+cosxsec2xtanx2

Conclusion

In this article we learned about differentiation of quotient of two functions, derivative of quotient of two functions, derivative of division of two functions, and differentiation of division of two functions which is done by using quotient rule of the differentiation. We also learned how to derive quotients by using different rules and properties of functions. In the end we also found how to use them with the help of examples.