## What does F Tell us about F?

What Does f ‘ Say About f? The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant. Because of this definition, the first derivative of a function tells us much about the function. If is positive, then must be increasing.

**How do you know if a function has a derivative?**

The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s no derivative — that happens in cases 1 and 2 below. In case 3, there’s a tangent line, but its slope and the derivative are undefined.

### What does a derivative do to a graph?

The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant. Because of this definition, the first derivative of a function tells us much about the function. If is positive, then must be increasing. If is negative, then must be decreasing.

**What does the second derivative tell you about a graph?**

The derivative tells us if the original function is increasing or decreasing. Because f′ is a function, we can take its derivative. The second derivative gives us a mathematical way to tell how the graph of a function is curved. The second derivative tells us if the original function is concave up or down.

#### How to find the derivative of a graph of a function?

Let’s begin with the fundamental connection between derivatives and graphs of functions. The derivative value f ‘ (a) equals the slope of the tangent line to the graph of y = f(x) at x = a. I recommend brushing up on the idea of tangent lines first. Here are a few resources that might help.

**What information does the graph of a function give?**

The graph of a function gives information about its derivative… if you know how to analyze it. The graph below shows the original in black and a sketch of its derivative in blue. Notice how the blue curve fits the description of f ‘. The blue curve is above the x -axis whenever f increases. The blue curve is below the x -axis whenever f decreases.

## How do you use the chain rule to find the derivative?

For instance, suppose we are given the following table of values for f, g, f’, and g’, and we want to find the instantaneous rate of change of h (x) at x = 1 given that h (x) = f (g (x)). See, we had to use the chain rule to calculate the derivative and then substitute the appropriate values from the table into the derivative and simplify.

**How do you find the derivative of a tangent line?**

The derivative value f ‘ (a) equals the slope of the tangent line to the graph of y = f(x) at x = a. I recommend brushing up on the idea of tangent lines first. Here are a few resources that might help. Is the Derivative of a Function the Tangent Line?