With Respect To (wrt) — Definition, Formula & Examples

'With respect to' (often abbreviated 'wrt') tells you which variable is changing or being acted on in a mathematical operation. For example, 'differentiate with respect to

x

' means treat

x

as the variable and everything else as a constant.

The phrase 'with respect to a variable $v$ ' specifies that $v$ is the independent variable of an operation—such as differentiation, integration, or solving—while all other symbols in the expression are treated as constants or parameters.

How It Works

When you see 'with respect to,' look for the variable named right after the phrase—that is the variable you focus on. In differentiation,

\frac{d}{dx}

means 'take the derivative with respect to

x

.' In integration, the

dx

at the end of

\int f(x)\,dx

serves the same purpose. If an expression contains multiple letters, this phrase removes ambiguity by telling you exactly which one varies.

Worked Example

Problem: Given

f(x, a) = 3a x^2 + 5x

, differentiate with respect to

x

Identify the variable: 'With respect to

x

' means

x

is the variable. Treat

a

as a constant.

Differentiate each term: Apply the power rule to each term, keeping

a

fixed.

\frac{d}{dx}(3a x^2) = 6ax, \quad \frac{d}{dx}(5x) = 5

Combine: Add the results.

\frac{df}{dx} = 6ax + 5

Answer:

\frac{df}{dx} = 6ax + 5

. If the problem had instead said 'with respect to

a

,' you would treat

x

as a constant and get

\frac{df}{da} = 3x^2

Why It Matters

In calculus and physics, expressions routinely contain several letters. Knowing which variable you differentiate or integrate 'with respect to' determines the entire answer. Misreading it is one of the fastest ways to get a problem completely wrong on an AP Calculus or university exam.

Common Mistakes

Mistake: Differentiating the wrong variable when multiple letters appear.

Correction: Always check what follows 'with respect to.' That variable gets the calculus treatment; every other letter acts as a constant.