The Quotient rule is generally used to find the derivative of a function. It is a formal rule for differentiating problems where one function gets divided by another function. The quotient rule states that the derivative of a quotient is the denominator (function written above the horizontal line) time the derivative of the numerator (function written below the horizontal line) minus the numerator times the derivative of denominator, all divided by square of the denominator.
Quotient Rule Formula
The quotient rule states that
if y = uv then:
dydx = v dudx – udvdx v²
Let’s learn how does above formula works:
Quotient Rule Example:
Differentiate cos x x²
Solution:
Let cos x = u
Let x² = v
Now, we will write the derivatives of these two functions:
dudx = -Sin x,
dvdx = 2x
Now, we will place these results into the formula:
dydx = v dudx – udvdx v²
dydx = x². (-sin x)- 2 cos x .2x (x²)²
Now, there is a minus sign (-) and y in both terms of the numerator. Here, we can consider -y as a common factor. Accordingly:
dydx = -x (x sinx)+ 2 cos x x⁴
= -x (xsiny)+ 2 cos x x³
By canceling the factor of x in both numerator and denominator. We have received the required derivative.
Chain Rule
What is Chain Rule?
The chain rule is used to find the derivative of a composite function. The chain rule enables us to use differentiation rules on more complex functions by differentiating both inner and outer functions separately.
Definition of Chain Rule
The chain rule in calculus is referred to as a formula for calculating the derivative of the composition of two or more functions. For example, if (f) and (g) are two functions, then the chain rule defines the derivative of the composite function f ∘ g in terms of the derivative of f and g.
What Does Chain Rule State?
The chain rule states that if there are two composite functions namely f(x) and g(x), then the composite function (f(g(x) is calculated for a value of (x) by first calculating g(x) and further determining the function (f) at this value for g(x), hence chaining the results together. For example:
If f(x) = Sin x , and g(x) = x²
Then,
(f(g(x) = (Sin x)²
In other words, the chain rule states that for a given composite function such as (f(g(x)Its derivative is the derivative of an outer function (without making any changes to the inner function and in place multiplied by the derivative of an inner function). Therefore, the chain rule is given as:
ddx f(g(x) = f'(g(x) . g'(x)
Let us understand chain rule with an example.
Evaluate the following: ddx = (3x + 1)²
Let g(x) = (3x + 1) and f(x)= x²
Accordingly, (3x + 1)² = f'(g(x)
As we know, g'(x)= 3 and f’ (x) = 3x. Hence,
ddx = (3x + 1)² = f'(g(x) . g'(x)
= f’ (3x + 1). 3
= 3(3x + 1). 3
= (9x + 3). 3
= 27 x + 9
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