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Casper Schorr
@casperschorr-1 Tasks: 6
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Joined: September 2024
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82611Released 10mo ago100% FreeDerivatives are a fundamental concept in calculus that measure the rate of change of a function. Here's a breakdown of what derivatives are and how they work: ### What is a derivative? A derivative is the instantaneous rate of change of a function with respect to one of its variables. It tells you how fast a function is changing at any given point. ### Key concepts: 1. **Slope**: The derivative at a point is equal to the slope of the tangent line to the function's graph at that point. 2. **Notation**: The derivative of a function f(x) is often written as f'(x) or dy/dx. 3. **Limit definition**: Formally, the derivative is defined as the limit of the difference quotient as the change in x approaches zero: f'(x) = lim[h→0] (f(x+h) - f(x)) / h ### Why are derivatives important? Derivatives have many practical applications, including: - Finding maximum and minimum values of functions - Optimization problems in economics, engineering, and physics - Analyzing rates of change in scientific data - Approximating functions with linear models ### Common derivative rules: 1. **Power rule**: (x^n)' = n * x^(n-1) 2. **Sum/difference rule**: (f(x) ± g(x))' = f'(x) ± g'(x) 3. **Product rule**: (f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x) 4. **Chain rule**: (f(g(x)))' = f'(g(x)) * g'(x) Understanding derivatives is crucial for many areas of mathematics, science, and engineering. They provide a powerful tool for analyzing changing quantities and optimizing functions.
