Basic Derivative
The derivative is a fundamental concept in calculus that measures the rate of change of a function at a particular point
The derivative is a fundamental concept in calculus that measures the rate of change of a function at a particular point. It gives us information about how the function is changing with respect to its input variable.
To find the derivative of a function, we use the rules of differentiation. These rules allow us to find the derivative of various types of functions. One of the basic rules is the power rule, which is used to find the derivative of a function raised to a constant power.
The power rule states that if we have a function f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is given by f'(x) = nx^(n-1).
Let’s illustrate this rule with an example:
Example:
Find the derivative of the function f(x) = 3x^2.
Solution:
To find the derivative, we can apply the power rule. According to the power rule, the derivative of f(x) = 3x^2 is given by f'(x) = 2 * 3 * x^(2-1) = 6x.
Therefore, the derivative of f(x) = 3x^2 is f'(x) = 6x.
This means that the rate of change of the function f(x) = 3x^2 is 6 times the value of x. In other words, as x increases, the function f(x) increases at a rate of 6 units for every unit increase in x.
It’s important to note that the power rule only applies when the exponent is a constant. It does not apply to functions with variable exponents, trigonometric functions, or logarithmic functions, among others. For those types of functions, different rules or techniques may be needed to find their derivatives.
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