Basic Derivative
A derivative is a fundamental concept in calculus that measures how a function changes as its input variable changes
A derivative is a fundamental concept in calculus that measures how a function changes as its input variable changes. It essentially tells us the rate of change of a function at a particular point.
To find the derivative of a function, you can follow a set of rules known as derivative rules. Here are some of the basic derivative rules:
1. Power Rule: If you have a function f(x) = x^n, where n is a constant, then its derivative f'(x) = n * x^(n-1). For example, if f(x) = x^2, its derivative is f'(x) = 2x.
2. Constant Rule: If you have a constant function f(x) = c, where c is a constant, then its derivative f'(x) = 0. For example, if f(x) = 5, its derivative is f'(x) = 0.
3. Sum/Difference Rule: If you have two functions f(x) and g(x), then the derivative of their sum or difference f(x) ± g(x) is the sum or difference of their derivatives, respectively. (f ± g)'(x) = f'(x) ± g'(x).
4. Product Rule: If you have two functions f(x) and g(x), then the derivative of their product f(x) * g(x) is given by the formula (f * g)'(x) = f'(x) * g(x) + f(x) * g'(x).
5. Quotient Rule: If you have two functions f(x) and g(x), then the derivative of their quotient f(x) / g(x) is given by the formula (f / g)'(x) = [f'(x) * g(x) – f(x) * g'(x)] / [g(x)]^2. Make sure the denominator is not zero.
These are just some of the basic derivative rules that you will encounter. Other rules, such as the chain rule for composite functions and the derivative of trigonometric functions, will come into play as you progress in calculus.
Remember that finding derivatives involves practice and understanding the rules, so don’t hesitate to work through examples and seek assistance if needed.
More Answers:
Using L’Hôpital’s Rule, we can determine the limit of sin(x)/x as x approaches 0, which is 1.Limit as x approaches 0: 1 – cos(x) / x = 1
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