How do you evaluate the derivative of a function when you don’t even know what the function is?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
That’s what finding derivatives using a table of values or graphs is all about, and it’s relatively straightforward!
How?
Because you are given everything you need in the form of a table or chart or graph, and all you have to do is plug in the numbers after applying a differentiation technique:
- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
What is important to note is that you are provided with select values that indicate various ordered pairs and an instantaneous rate of change for specified x-values for the given functions.
More importantly, these functions are continuous and differentiable, meaning we will be able to calculate their derivatives from the given table of values at select points along each curve, and not have to worry about our calculations becoming undefined.
Example
For instance, suppose we are given the following table of values for f, g, f’, and g’, and we want to find the instantaneous rate of change of h(x) at x = 1 given that h(x) = f(g(x)).
Find Derivatives Using Table of Values
See, we had to use the chain rule to calculate the derivative and then substitute the appropriate values from the table into the derivative and simplify.
Phew!
But why do we need to know how to find derivatives using a table of values?
Well, we encounter “table of values” questions whenever we are given data points and are asked to estimate the rate of change of things like:
- Interest Rates
- Unemployment Rates
- Marginal Cost
- Population Growth
- Inverse Functions
- Linearization
And, sometimes we are only given a graph, and we must use our understanding of differentiability to find the instantaneous and average rate of change.
Example
For this problem, use the graph of f’ as seen below, estimate the value of f’ (-5), f’ (-3), f’ (-1), and f’ (0).
Graph – Continuous Function
All we have to do is estimate the slope of the tangent line (i.e., the instantaneous rate of change) at each of the specified x-values.
Find The Slope Line Tangent At Point Using A Graph
See, that wasn’t so scary.
So, together, we will review the power, product, quotient, and chain rules within our five examples and see how to find the instantaneous rate of change of a function even when the curve is not explicitly provided.
Video Tutorial w/ Full Lesson & Detailed Examples (Video)
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