Introduction to derivatives
A derivative is a way to measure how a function changes as it’s input changes. In other words a derivative tell us how the rate of change of a given function. The derivative is the ratio of the infinitesimal change of the output over the infinitesimal change of the input producing that change of output.
The definition of a derivative is the instantaneous rate of change formula.
The definition of a derivative is the instantaneous rate of change formula.
The instantaneous rate of change formula is saying that the derivative of f(x) with respect to x (ddx) is the function f’(x).
Now we can solve some example problems.
Now we can solve some example problems.
Example Problems
1) Find there derivative of the function f(x) with respect to x.
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In addition to finding instantaneous rate of change, we can also calculate the instantaneous rate of change at a point. This ROC change will be the slope of the tangent line. We can also find the average rate of change when given two points. This is slope of the secant line. To find the tangent line we can use the difference quotient(similar to the instantaneous rate of change without the limit), and to find the secant line we can use the slope.
The following slope formula can be used to calculate the slope of the secant line connecting a and b.
Now we can solve some example problems.
Example Problems
TANGENT LINE: Find the equation of the tangent line of the function f(x) at the point when x=3.
Use the instantaneous rate of change formula that we used earlier to find the derivative.
Now substitute 3 for x.
25 is the slope of the tangent line at x=3. Now use the equation of a linear function to find the equation of the tangent line.
Find the (x,y) coordinate to solve for b. We already have the coordinate x, now plug in 3 into the original f(x) function to find y.
Now that we have the coordinates we can find the equation of the tangent line.