Find the slope of a line tangent to a function at a certain point.

Understand secant and tangent lines.

Derive the formula for a derivative and use the formula to find slope.

Step 1:

You are given a point of tangency where the tangent line meets the curve.

Ex: Find the slope of the line tangent to 2x3 at

(1,2). This will be your first point for calculating slope.

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Step 2:

You have to find another point on the line. To do this, add a number (h) to the x value.

Calculate the corresponding y value using the function.

You now have a point (x+h, f(x+h)) . For accuracy, you want h to be as small as possible.

Step 2:

By doing this, you are actually calculating the slope of the secant line (passes through at

two points, rather than only touching once).

The goal is to get h as small as possible so the secant line is as close to the tangent line as possible (your estimation is as close to the actual value as possible).

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Step 3:

Calculate the slope by (y1 – y2)/(x1 – x2).

You now have an estimation of the slope of the tangent line.

But how do you get h to be as small as

possible? You want to find the limit of h -> 0.

Formula:

h

xfhxf

h

)()(lim

0

Use limits to get h ASAP

Find y value of x+h

Subtract to f ind slope

Y value of original point

Difference in x values = h

Step 4:

Find the equation of the line by plugging in the original point and the slope.

Example 1:

Find the equation of the line tangent to x2 + 3 at (2,7).

Step 1:

The derivative follows the formula for the slope of the tangent line, but doesn’t plug in

specific points. It is the general equation for the slope of the

tangent line at any point. Fill in f(x) with your equation.

Step 2:

Once your equation is as simple as possible, take the limit.

This will give you an equation. You can plug in

x to find the slope of the tangent line at any point on the curve.

Example 2:

Find the derivative of 2x2 + 5. Use this to find the slope at (1,7).