To find the slope of a curve at a certain point, we use something called derivatives!
The derivative tells us how fast a function is changing. Imagine it's like finding the slope of a straight line that just touches the curve at that point.
Let’s Break It Down:
What is a Derivative?
The derivative of a function, which we can write as ( f'(x) ) or ( \frac{df}{dx} ), shows how the function ( f(x) ) changes when ( x ) changes.
Finding the Slope:
To get the slope at a specific point (let’s say when ( x = a )), we just find the derivative at that point: ( f'(a) ).
Example:
Let’s look at the function ( f(x) = x^2 ). If we want to find the slope when ( x = 3 ):
First, we find the derivative:
( f'(x) = 2x )
Now, we plug in ( x = 3 ):
( f'(3) = 2(3) = 6 )
So, the slope of the curve at the point (3, 9) is 6!
Illustration:
Imagine you graph the function ( f(x) = x^2 ). Draw a line that just touches the curve at the point (3, 9).
The slope of this line is exactly what we just calculated with the derivative!
Using derivatives is a great way to understand how functions work at specific points.
To find the slope of a curve at a certain point, we use something called derivatives!
The derivative tells us how fast a function is changing. Imagine it's like finding the slope of a straight line that just touches the curve at that point.
Let’s Break It Down:
What is a Derivative?
The derivative of a function, which we can write as ( f'(x) ) or ( \frac{df}{dx} ), shows how the function ( f(x) ) changes when ( x ) changes.
Finding the Slope:
To get the slope at a specific point (let’s say when ( x = a )), we just find the derivative at that point: ( f'(a) ).
Example:
Let’s look at the function ( f(x) = x^2 ). If we want to find the slope when ( x = 3 ):
First, we find the derivative:
( f'(x) = 2x )
Now, we plug in ( x = 3 ):
( f'(3) = 2(3) = 6 )
So, the slope of the curve at the point (3, 9) is 6!
Illustration:
Imagine you graph the function ( f(x) = x^2 ). Draw a line that just touches the curve at the point (3, 9).
The slope of this line is exactly what we just calculated with the derivative!
Using derivatives is a great way to understand how functions work at specific points.