How to compute the derivative of a function (in R)

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Task

Given a mathematical function $f(x)$, we write $f^{\prime }(x)$ or $\frac{d}{dx}f(x)$ to represent its derivative, or the rate of change of $f$ with respect to $x$. How can we compute $f^{\prime }(x)$ using mathematical software?

Solution

Let’s consider the function $f(x)=10x^2-16x+1$. We focus not on the whole function, but just the expression on the right-hand side, $10x^2-16x+1$.

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formula <- expression( 10*x^2 - 16*x + 1 )

We can compute its derivative using the D function.

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D( formula, "x" )   # derivative with respect to x

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10 * (2 * x) - 16

R does not simplify the output, but if we do so ourselves, we find that $f^{\prime }(x)=20x-16$.

If it had been a multi-variable function, we would need to specify the variable with respect to which we wanted to compute a derivative.

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formula2 <- expression( x^2-y^2 )  # consider the formula x^2 - y^2
D( formula2, "y" )                 # differentiate with respect to y

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-(2 * y)

That output says that $\frac{\partial }{\partial y}(x^2-y^2)=-2y$.

We can compute the second derivative by using the D function twice and specifying the variables with respect to which we are computing the derivative.

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D( D( formula2, "x" ), "x" )  # second derivative with respect to x

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[1] 2

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D( D( formula2, "x" ), "y" )  # mixed partial derivative

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[1] 0

Content last modified on 24 July 2023.

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Contributed by Debayan Sen (DSEN@bentley.edu)