The slope of the tangent to the graph of a function at x=x₀ can be estimated by finding the slope of a line that passes through two points near to x₀.
For instance, evaluate the function f at x₀+h and x₀. The slope will be given by:
For small values of h, this slope will approach the slope of the tangent line.
For the given function:
Use x₀=2 and h=0.01. Then, we must evaluate f at x=2.01 and at x=2:
Then, the slope of the tangent line is approximately:
We can get better approximations by choosing smaller values for h. For instance, if h=0.001, we would get:
Therefore, the slope of the line tangent to the graph of y=2x³+x²+23 is approximately 28.