© 2025 Astra.si. All rights reserved.
"For the next generation"
© 2025 Astra.si. All rights reserved.
"For the next generation"
Derivatives are a fundamental concept in mathematics, particularly in differential calculus, that describes how the value of a function changes as its input values change. In other words, the derivative of a function at a given point measures the slope of the tangent line to the graph of the function at that point and is central to understanding the dynamics of change.
Mathematically, the derivative of a function 'f' with respect to a variable 'x' is expressed as the limit of the difference quotient of the function as the difference 'h' between two values of 'x' approaches zero. (f'(x) = lim (as h→0) [f(x+h) - f(x)]/h).
The slope of the tangent to the graph of a function at point 'x' is actually the derivative of the function at that point. If we draw a tangent line to the graph of the function at point 'x', the slope (or steepness) of this tangent will be equal to the value of the derivative f'(x) at point 'x'.
Imagine you are graphing a function, for example, f(x) = x². When you draw a tangent line to the graph of this function at a specific point, for instance, at x = 3, this line will have a certain slope. This slope of the line is the value of the derivative f'(x) at point x = 3. For our example, f'(x) = 2x, which means that f'(3) = 6. This tells us that the slope of the tangent to the graph of f(x) = x² at the point x = 3 is 6.
Let's take the function f(x) = x². Its derivative is f'(x) = 2x. If we want to find the slope of the tangent at the point x = 2, we simply calculate the derivative at that point: f'(2) = 2 * 2 = 4. Therefore, the slope of the tangent to the graph of the function f(x) = x² at the point x = 2 is 4.
