Understanding Derivatives
Derivatives are a fundamental tool in calculus, representing the rate of change of a function. They are essential in fields ranging from physics to finance.
In this guide, we'll explore the basics of derivatives and how to calculate them. Derivatives can be understood both geometrically and algebraically.
Geometric Interpretation
Geometrically, the derivative of a function at a point is the slope of the tangent line to the function's curve at that point.
Basic Rules of Differentiation
- Power Rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
- Sum Rule: The derivative of the sum of two functions is the sum of their derivatives.
- Product Rule: If \( f(x) \) and \( g(x) \) are functions, then \( (f \cdot g)' = f'g + fg' \).
- Quotient Rule: If \( f(x) \) and \( g(x) \) are functions, then \( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} \).
Example Problem
Let's differentiate \( f(x) = 3x^3 + 2x^2 - x + 7 \).
Using the power rule, we find: \( f'(x) = 9x^2 + 4x - 1 \).
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