Quotient Rule Formula (original) (raw)
Last Updated : 23 Jul, 2025
The quotient rule is a fundamental technique in calculus used to differentiate functions that are expressed as the ratio of two other functions. It's especially useful when dealing with more complex expressions that cannot be simplified easily before differentiation.
**Quotient Rule in Calculus
In calculus, the quotient rule is a technique for determining the derivative of any function provided in the form of a quotient derived by dividing two differentiable functions. The quotient rule says that the derivative of a quotient is equal to the ratio of the result achieved by subtracting the numerator times the denominator's derivative from the denominator times the numerator's derivative to the square of the denominator's derivative.
If we have a function of the type u(x)/v(x), we can use the quotient rule derivative to obtain the derivative of that function. The formula for the quotient rule is as follows:
\frac{d\left(\frac{u(x)}{v(x)}\right)}{dx}=[v(x) × u'(x) - u(x) × v'(x)]/[v(x)]^2
where,
- u(x) and v(x) are differentiable functions in R.
- u'(x) and v'(x) are the derivatives of functions u(x) and v(x) respectively.
**Derivation
Suppose a function f(x) = u(x)/v(x) is differentiable at x. We will prove the product rule formula using the definition of derivative or limits.
f'(x)=\lim_{\Delta x\to0} \frac{f(x+\Delta x)-f(x)}{\Delta x}
= \lim_{\Delta x\to0} \frac{\frac{u(x+\Delta x)}{v(x+\Delta x)}-\frac{u(x)}{v(x)}}{\Delta x}
= \lim_{\Delta x\to0}\frac{u(x+\Delta x)v(x)-u(x)v(x+\Delta x)}{v(x+\Delta x)v(x)\Delta x}
= \frac{1}{[v(x)]^2}\lim_{\Delta x\to0}\frac{u(x+\Delta x)v(x)-u(x)v(x+\Delta x)}{\Delta x}
= \frac{1}{[v(x)]^2}\lim_{\Delta x\to0}\frac{u(x+\Delta x)v(x)-u(x)v(x)+u(x)v(x)-u(x)v(x+\Delta x)}{\Delta x}
= \frac{1}{[v(x)]^2}\lim_{\Delta x\to0}\frac{v(x)(u(x+\Delta x)-u(x))-u(x)(v(x+\Delta x)-v(x))}{\Delta x}
= \frac{1}{[v(x)]^2}\left[v(x)\lim_{\Delta x\to0}\frac{u(x+\Delta x)-u(x)}{\Delta x}-u(x)\lim_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}\right]
Put \lim_{\Delta x\to0} \frac{u(x+\Delta x)-u(x)}{\Delta x}=u'(x) and \lim_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}=v'(x)
= [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
This derives the formula for quotient rule.
**Sample Problems
**Question 1. Find the derivative of the function f(x) = 1/x using quotient rule.
**Solution:
We have, f(x) = 1/x. Here, u(x) = 1 and v(x) = x.
So, u'(x) = 0 and v'(x) = 1
Using quotient rule we have,
f'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
= [x (0) - 1 (1)/x2
= 1/x2
**Question 2. Find the derivative of the function f(x) = 1/sin x using quotient rule.
**Solution:
We have, f(x) = 1/sin x. Here, u(x) = 1 and v(x) = sin x.
So, u'(x) = 0 and v'(x) = cos x
Using quotient rule we have,
f'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
= [sin x (0) - 1 (cos x)]/cos2 x
= -cos x/ cos2 x
= -1/cos x
= -sec x
**Question 3. Find the derivative of the function f(x) = x/sin x using quotient rule.
**Solution:
We have, f(x) = x/sin x. Here, u(x) = x and v(x) = sin x.
So, u'(x) = 1 and v'(x) = cos x
Using quotient rule we have,
f'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
= [sin x (1) - x (cos x)]/cos2 x
= (sin x - x cos x)/cos2 x
**Question 4. Find the derivative of the function f(x) = cos x/x using quotient rule.
**Solution:
We have, f(x) = cos x/x. Here, u(x) = cos x and v(x) = x.
So, u'(x) = -sin x and v'(x) = 1
Using quotient rule we have,
f'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
= [x (-sin x) - cos x (1)]/x2
= (-x sin x - cos x)/x2
**Question 5. Find the derivative of the function f(x) = log x/x using quotient rule.
**Solution:
We have, f(x) = log x/x. Here, u(x) = log x and v(x) = x.
So, u'(x) = 1/x and v'(x) = 1
Using quotient rule we have,
f'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
= [x (1/x) - log x (1)]/x2
= (1 - log x)/x2
**Question 6. Find the derivative of the function f(x) = (2x - 1)/x 2 using quotient rule.
**Solution:
We have, f(x) = (2x - 1)/x2. Here, u(x) = 2x - 1 and v(x) = x2.
So, u'(x) = 2 and v'(x) = 2x
Using quotient rule we have,
f'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
= [x2 (2) - (2x - 1) (2x)]/x4
= (2x2 - 4x2 + 2x)/x4
= (-2x2 + 2x)/x4
= [-2x(x - 1)]/x4
= -2(x - 1)/x3
**Question 7. Find the derivative of the function f(x) = log x/sin x using quotient rule.
**Solution:
We have, f(x) = log x/sin x. Here, u(x) = log x and v(x) = sin x.
So, u'(x) = 1/x and v'(x) = cos x
Using quotient rule we have,
f'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
= [sin x (1/x) - log x (cos x)]/sin2 x
= [sin x/x - log x cos x]/sin2 x
Practice Problems on Quotient Rule Formula
**Question 1: Differentiate f(x) = \frac{2x³+5x}{x²+3} using Quotient rule
**Question 2: Find the derivative of f(x) = \frac{ln(x)}{x²}
**Question 3: Use the quotient rule to differentiate f(x) = \frac{e^x}{x+1}
**Question 4: Find the derivative of f(x) = \frac{5x²-4x + 1}{2x^3-x}
Conclusion
The quotient rule is a powerful tool in calculus that simplifies the differentiation of ratios of functions. While it may seem complex at first, with practice, it becomes an intuitive method for most differentiation problems. Understanding the derivation, application, and potential of the quotient rule is essential for mastering calculus and its applications in various disciplines.
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