R Lesson 28: Another Way To Work With Derivatives in R

Hello everybody,

Michael here, and in today’s lesson, I’ll show you another cool way to work with derivatives in R.

In the previous post-R Lesson 27: Introductory R Calculus-I discussed how to work with derivatives in R. However, the derivatives method I discussed in that post doesn’t cover the built-in derivatives method R has to calculate derivatives…rather, methods R uses to calculate derivatives (there are two ways of approaching this). What might that method look like? Well, lets dive in!

Calculating derivatives, the built-in R way, part 1

Now, how can we calculate derivatives with the built-in R way? First, we’ll explore the deriv() function. Let’s take a look at this code below, which contains a simple equation for a parabolic curve:

curve <- expression(3*x^2+4*x+5)
print(deriv(curve, "x"))

expression({
    .value <- 3 * x^2 + 4 * x + 5
    .grad <- array(0, c(length(.value), 1L), list(NULL, c("x")))
    .grad[, "x"] <- 3 * (2 * x) + 4
    attr(.value, "gradient") <- .grad
    .value
})

In this example, I’m using the polynomial 3x^2+4x+5 to represent our hypothetical parabolic curve. To find the derivative function of this polynomial, I ran R’s built in deriv() function and passed in both the curve expression and "x" (yes, in double quotes) to find the derivative expression-as the derivative always relates to x (or whatever character you used to represent an unknown quantity). Now, as you see from the output given, things don’t look too understandable. However, pay attention to the second .grad line (in this case, 3 * (2 * x) + 4), as this will provide the derivative function of the parabolic curve equation-6x+4. We would use this equation to evaluate the rate of change between points in the parabolic curve.

Let’s say we wanted to evaluate the derivative polynomial-6x+4-at x=5. If we use this x value, then the derivative of the parabola at x=5 would be 34.

• If you thought the output I’m referring to read as 3(2x)+4, you’d be wrong. Remember to multiply the 3 by the 2x to get the correct answer of 6x (6x+4 to be exact).
• When you’re writing a polynomial in R, remember to include all the multiplication signs (*)-yea, I know it’s super annoying.

In case you wanted a visual representation of this parabola, here it is:

I created this illustation of a parabola using a free online tool called DESMOS, which allows you to quickly create a visual representation of a line, parabola, or other curve. Here’s the link to DESMOS-https://www.desmos.com/calculator/dz0kvw0qjg.

Calculating derivatives, the built-in R way, part 2

Now let’s explore R’s other built-in way to calculate derivatives-this time using the D() function. For this example, let’s use the same parabolic curve equation we used for the deriv() example:

curve <- expression(3*x^2+4*x+5)
print(D(curve, "x"))

3 * (2 * x) + 4

As you can see, we passed in the same parameters for the D() function that we used for the deriv() function and got a much simpler version of the output we got for the deriv() function-simpler as in the derivative expression itself was the only thing that was returned.

Since the derivative expression returned from the D() function is the same as the expression returned from the deriv() function,

Calculating derivatives, the manual way, part 3

Yes, I know I was mostly going to focus on the two built-in R functions used to calculate derivatives-deriv() and D()-but I thought I’d include this bonus section for those of you (especially those who enjoy exploring calculus) who were wondering how to manually calculate derivatives.

Let’s take the same polynomial we were working with for the previous two examples:

How would we arrive at the derivative expression of 6x+4? Check out this illustration below:

From this picture, here are some things to keep in mind when calculating derivatives of polynomials (and it’s not the same as calculating derivatives of regular numbers like we did in the previous post R Lesson 27: Introductory R Calculus):

• Go term-by-term when calculating derivatives of polynomials. In this example, you’d calculate the derivative of 3x^2, then the derivative of 4x and lastly the derivative of 5.
• How would you calculate the derivatives of each term? Here’s how (and it’s quite easy).
• The derivative of 3x^2 would be 6x, as you would multiply the number (3) by the power (2) to get 6. You would then reduce power of the x^2 by 1 to get x (any variable in a polynomial without an exponent is raised to the power of 1). Thus, the derivative of 3x^2 would be 6x.
• The derivative of 4x would simply be 4. Just as we did with 3x^2, we’d multiply the number (4) by the power (1) in this case to get 4. We would also reduce the power of x by 1 to simply get 4, since x has a power of 1 and 1-1=0. In a polynomial, constants (numbers without variables next to them) have a power of 0.
• I mean, we could’ve written the derivative polynomial as 6x+4x^0, but 6x+4 looks a lot nicer.
• As for the constant in this polynomial-5-it has a derivative of 0 since the derivative of a constant is always 0 (after all, any constant in a polynomial has a power of 0, so this makes perfect sense). Thus, the derivative of 5 isn’t included in the derivative polynomial of 6x+4.

Thanks for reading,

Michael