We can use differentiation to approximate the value of certain functions at different points. Recall that the limit definition of a derivative gives us an expression for the derivative of a function using limits. If I eliminate the limit in this definition, I can state an approximation for the derivative of the function as follows:
Using the second expression, we can obtain certain approximations by rearranging it as:
Consider the square root of 25.2. If we want to approximate this value, we can use the previous expression with x=25, h=0.2 and f(x) = x^(1/2). Then, we find:
Differentials are a minor concept but can be useful when you do not have a calculator around and need a numerical expression for a certain problem. There is an alternative method to the one described thus far. Instead of using the approximation for f(x+h), one can write an equation for the tangent line of the curve at the point x and then plug in x+h into the equation for the tangent line. Here’s the same example using this alternative method:
To write a tangent line to the curve f(x) = x^(1/2) at (25,5), we must find the slope of the curve at x=25. Taking the derivative and then plugging in x=25, we find f'(25) = 1/10. Then, using point-slope form, we have
Finally, we plug in x=25.2. Doing so yields the same value we obtained using the other method: y=5.02. You should choose the method that is simplest based on the curve.
There will be some error in the approximation but its simplicity makes it appealing.