Mastering Differentiation: The Power, Product, Quotient, and Chain Rules for Calculating Derivatives

Basic Derivative

The derivative of a function measures how the function changes as its input variable changes

The derivative of a function measures how the function changes as its input variable changes. It is denoted by prime notation (‘) or by using the \(\frac{d}{dx}\) operator. The process of finding the derivative of a function is called differentiation.

To differentiate a function, you need to follow a set of rules, including the power rule, product rule, quotient rule, and chain rule.

1. Power Rule:
For a function \(f(x) = x^n\), where \(n\) is a constant exponent, the derivative is given by \(f'(x) = nx^{n-1}\). For example, the derivative of \(f(x) = x^2\) is \(f'(x) = 2x\).

2. Product Rule:
For two functions \(f(x)\) and \(g(x)\), the derivative of their product \(h(x) = f(x) \cdot g(x)\) is given by \(h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)\). For example, if \(f(x) = x^2\) and \(g(x) = x\), then \(h(x) = f(x) \cdot g(x) = x^3\) and \(h'(x) = 2x \cdot x + x^2 \cdot 1 = 3x^2\).

3. Quotient Rule:
For two functions \(f(x)\) and \(g(x)\), the derivative of their quotient \(h(x) = \frac{f(x)}{g(x)}\) is given by \(h'(x) = \frac{f'(x) \cdot g(x) – f(x) \cdot g'(x)}{(g(x))^2}\). For example, if \(f(x) = x^2\) and \(g(x) = x\), then \(h(x) = \frac{f(x)}{g(x)} = \frac{x^2}{x} = x\) and \(h'(x) = \frac{2x \cdot x – x^2 \cdot 1}{x^2} = 1\).

4. Chain Rule:
If a function \(f(x)\) is composed with another function \(g(x)\) such that \(f(x) = g(h(x))\), then the chain rule states that the derivative of \(f(x)\) is given by \(f'(x) = g'(h(x)) \cdot h'(x)\). For example, if \(f(x) = (2x)^3\), then \(g(x) = x^3\) and \(h(x) = 2x\), so \(f(x) = g(h(x))\) and \(f'(x) = g'(h(x)) \cdot h'(x)\). To find \(f'(x)\), we first differentiate \(g(x) = x^3\) to get \(g'(x) = 3x^2\), and then differentiate \(h(x) = 2x\) to get \(h'(x) = 2\). Finally, substituting these values into the chain rule formula, we have \(f'(x) = g'(h(x)) \cdot h'(x) = 3(2x)^2 \cdot 2 = 12x^2\).

By using these rules, you can find the derivative of more complex functions. It is also important to note that the derivative gives us information about the slope of a function at any given point and can be used to find maximum or minimum points of a function.

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