Derivatives give you the rate of change (or slope) at a single point on a line. The line can be curved or straight -- it doesn't matter. Think of it as how much the line is changing at any time, instead of the slope of the entire line. How you take derivatives changes depending on the type of function you have, so review how to take common derivatives before moving on.


Review taking derivatives here

The most simple derivatives, those for basic exponential equations, are easy to find using a simple shortcut. This will be used for the rest of the method.
 You will not always be asked to explicitly find the derivative or slope of a curve. You might also be asked for the "rate of change at point (x,y). You could be asked for an equation for the slope of the graph, which simply means you need to take the derivative. Finally, you may be asked for "the slope of the tangent line at (x,y)." This, once again, just wants the slope of the curve at a specific point, (x,y).

For this method, consider the question: "What is the slope of the line f(x)=2x2+6x{\displaystyle f(x)=2x^{2}+6x} at the point (4,2)?"The derivative is often written as f′(x),y′,{\displaystyle f'(x),y',} or dydx{\displaystyle {\frac {dy}{dx}}}, You don't even really need you graph, just the function or equation for your graph. For this example, use the function from earlier, f(x)=2x2+6x{\displaystyle f(x)=2x^{2}+6x}. Following the methods outlined here, take the derivative of this simple function.


Derivative: f′(x)=4x+6{\displaystyle f'(x)=4x+6}


, The differential of a function will tell you the slope of the function at a given point. In other words, f’(x) is slope of the function at any point (x,f(x)) So, for the practice problem:

What is the slope of the line f(x)=2x2+6x{\displaystyle f(x)=2x^{2}+6x} at the point (4,2)?


Derivative of Equation:


f′(x)=4x+6{\displaystyle f'(x)=4x+6}




Plug in Point for x:


f′(x)=4(4)+6{\displaystyle f'(x)=4(4)+6}




Find the Slope:

The slope of the f(x)=2x2+6x{\displaystyle f(x)=2x^{2}+6x} at (4,2) is 22.
 Know that not all points in calculus will have a slope. Calculus gets into complex equations and difficult graphs, and not all points will have a slope, or even exist on every graph. Whenever possible, use a graphing calculator to check the slope of your graph. If you can't, draw the tangent line using your point and the slope (remember -- "rise over run") and note if it looks like it could be correct.

Tangent lines are just lines with the exact same slope as your point on the curve. To draw one, go up (positive) or down (negative) your slope (in the case of the example, 22 points up). Then move over one and draw a point. Connect the dots, (4,2) and (26,3) for your line.
Review how to take a variety of derivatives from common functions.
Understand what questions are asking for a slope using derivatives.
Take the derivative of your function.
Plug in your point to the derivative equation to get your slope.
Check your point against a graph whenever possible.
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