Derivative – Exercise 2

Video explanation

Explanation

The derivative of a function is a key tool in differential calculus that allows for determining the rate of change of a function with respect to its variable. This process reveals fundamental properties of functions, such as the slopes of tangents, extrema (maximums and minimums), and inflection points. The following describes how to calculate the derivatives of a function f(x) and what the key aspects are.

BASIC CONCEPTS

The derivative of a function f(x) at a point x is defined as the limit of the ratio of the change in the function at that point as the interval of change, h, approaches 0. This is mathematically expressed as: f′(x) = lim (as h→0) [f(x+h) – f(x)] / h

RULES FOR DIFFERENTIATION

DERIVATIVE OF A CONSTANT: The derivative of any constant is 0.
DERIVATIVE OF A POWER FUNCTION: The derivative of x^n is nx^(n-1).
PRODUCT RULE: The derivative of the product of two functions, f(x)g(x), is f′(x)g(x) + f(x)g′(x).
QUOTIENT RULE: The derivative of the quotient of two functions, f(x)/g(x), is [f′(x)g(x) – f(x)g′(x)] / (g(x))².
CHAIN RULE: Used when differentiating composite functions.

EXAMPLES

For a practical example, if we want to calculate the derivative of the function f(x) = x², we use the power rule to get f'(x) = 2x.

USES OF DERIVATIVES

Derivatives allow for the determination of the rate of change of a function, which is useful in finding maximums, minimums, and analyzing the tangent to a curve. Derivatives are also the foundation for further concepts in mathematics, such as integrals and differential equations.

CONCLUSION

Knowledge of how to calculate the derivatives of a function is fundamental for the study and understanding of natural phenomena and technical applications. Understanding and using the basic rules of differentiation allows for deeper insights into the properties of functions and is indispensable when solving mathematical problems. Understanding derivatives opens the door to comprehending more complex concepts in mathematics and their practical application.

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Definition of the Derivative
Stationary Points, Increasing/Decreasing Functions
Slope of the Tangent and Normal Line
Derivatives of Other Elementary Functions