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FAQs on Derivative Rules - Business Mathematics and Statistics - B Com
| 1. What are the basic rules of derivatives in calculus? |  |
Ans. The basic rules of derivatives include the power rule, product rule, quotient rule, and chain rule. The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). The product rule states that the derivative of two functions multiplied together is \(f'g + fg'\). The quotient rule states that the derivative of a function divided by another is \(\frac{f'g - fg'}{g^2}\). The chain rule is used to differentiate composite functions, and it states that the derivative of \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\).
| 2. How do you apply the chain rule in differentiation? | |
Ans. To apply the chain rule, identify the outer function and the inner function in a composite function. For example, if you have \(y = f(g(x))\), you first differentiate the outer function \(f\) with respect to \(g(x)\), then multiply by the derivative of the inner function \(g\) with respect to \(x\). This is expressed mathematically as \(y' = f'(g(x)) \cdot g'(x)\).
| 3. What is the product rule and when should it be used? | |
Ans. The product rule is used when you need to differentiate a product of two functions. It states that if \(y = u \cdot v\), where \(u\) and \(v\) are functions of \(x\), then the derivative \(y'\) is given by \(y' = u'v + uv'\), where \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\), respectively. This rule is essential when both functions are multiplied together and you need to find the rate of change of their product.
| 4. Can you explain the quotient rule with an example? | |
Ans. The quotient rule is used when differentiating a quotient of two functions. If \(y = \frac{u}{v}\), the derivative is given by \(y' = \frac{u'v - uv'}{v^2}\). For example, if \(y = \frac{x^2}{x+1}\), then \(u = x^2\) and \(v = x + 1\). The derivatives are \(u' = 2x\) and \(v' = 1\). Applying the quotient rule, we get \(y' = \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2}\).
| 5. What is the significance of the first derivative in business applications? | |
Ans. The first derivative is significant in business as it represents the rate of change of a function, which can indicate trends in revenue, cost, or profit. For example, if a company wants to understand how changes in production levels affect profit, the first derivative of the profit function with respect to production can help identify maximum profit points and optimal production levels, guiding decision-making and strategy.
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Sep 21, 2025
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