Basic Derivative
The derivative is a fundamental concept in calculus that measures the rate at which a function changes at a specific point
The derivative is a fundamental concept in calculus that measures the rate at which a function changes at a specific point. It represents the slope of the tangent line to the graph of the function at that point.
To find the derivative of a function, you can use the power rule, which states that for any real number n and a constant a:
d/dx (a * x^n) = n * a * x^(n-1)
Here are some examples to illustrate how to use this rule:
Example 1: Find the derivative of f(x) = 3x^2 + 5
Using the power rule, differentiate each term:
d/dx (3x^2) = 2 * 3 * x^(2-1) = 6x
d/dx (5) = 0 (since the derivative of a constant is always zero)
So, the derivative of f(x) = 3x^2 + 5 is f'(x) = 6x.
Example 2: Find the derivative of g(x) = 4x^3 – 2x^2 + 7x
Again, using the power rule, differentiate each term:
d/dx (4x^3) = 3 * 4 * x^(3-1) = 12x^2
d/dx (-2x^2) = 2 * -2 * x^(2-1) = -4x
d/dx (7x) = 1 * 7 * x^(1-1) = 7
So, the derivative of g(x) = 4x^3 – 2x^2 + 7x is g'(x) = 12x^2 – 4x + 7.
These examples demonstrate the basic concept of finding the derivative using the power rule. However, there are several other rules and techniques in calculus for finding derivatives, such as the product rule, quotient rule, and chain rule, which are used for more complex functions.
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