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Differential

Differential

An tiny or infinitesimal change in the value of a variable. Differentials are commonly written in the form dx or dy.

 

 

See also

Approximation by differentials

Key Formula

dy=f′(x) dxdy = f'(x)\,dx
Where:
  • dydy = The differential of y — the approximate change in y
  • f′(x)f'(x) = The derivative of the function y = f(x)
  • dxdx = The differential of x — a small change in the independent variable x

Worked Example

Problem: Given y = x², use differentials to approximate the change in y when x changes from 3 to 3.01.
Step 1: Identify the function and find its derivative.
y=x2  ⟹  f′(x)=2xy = x^2 \implies f'(x) = 2x
Step 2: Set up the values. Here x = 3 and the small change in x is dx = 0.01.
x=3,dx=0.01x = 3, \quad dx = 0.01
Step 3: Apply the differential formula dy = f'(x) dx.
dy=2(3)(0.01)=0.06dy = 2(3)(0.01) = 0.06
Step 4: Compare with the exact change: Δy = (3.01)² − 3² = 9.0601 − 9 = 0.0601. The differential dy = 0.06 is very close to the actual change.
Δy=0.0601,dy=0.06\Delta y = 0.0601, \quad dy = 0.06
Answer: The differential dy = 0.06, which approximates the actual change of 0.0601 with high accuracy.

Another Example

Problem: Use differentials to approximate √16.2 without a calculator.
Step 1: Choose a nearby value where the square root is exact. Let f(x) = √x, with x = 16 and dx = 0.2.
f(x)=x,x=16,dx=0.2f(x) = \sqrt{x}, \quad x = 16, \quad dx = 0.2
Step 2: Find the derivative of f(x).
f′(x)=12xf'(x) = \frac{1}{2\sqrt{x}}
Step 3: Compute the differential dy.
dy=1216⋅0.2=0.28=0.025dy = \frac{1}{2\sqrt{16}} \cdot 0.2 = \frac{0.2}{8} = 0.025
Step 4: Add the differential to f(16) to get the approximation.
16.2≈4+0.025=4.025\sqrt{16.2} \approx 4 + 0.025 = 4.025
Answer: √16.2 ≈ 4.025 (the actual value is 4.02492..., so the approximation is excellent).

Frequently Asked Questions

What is the difference between dx and Δx?
Both represent a change in x, but Δx denotes a finite, measurable change of any size, while dx represents an infinitesimally small change — a change that is conceptually approaching zero. When you use dx in the differential formula dy = f'(x) dx, you treat dx as arbitrarily small so the linear approximation is extremely accurate. In practice for computations, you often substitute a small Δx value in place of dx.
Why does dy/dx look like a fraction if it's a derivative?
The notation dy/dx originated from the ratio of two differentials: dy and dx. While the derivative is formally defined as a limit, Leibniz's notation preserves the idea that it behaves like a fraction of infinitesimal changes. This is why you can "multiply both sides by dx" to get dy = f'(x) dx — the algebraic manipulation is valid because differentials are well-defined quantities.

Differential (dy) vs. Derivative (dy/dx)

A differential dydy is a quantity — it tells you how much yy changes for a given small change dxdx. A derivative dy/dxdy/dx is a rate — it tells you the ratio of change in yy to change in xx at a point. The relationship is dy=dydx⋅dxdy = \frac{dy}{dx} \cdot dx. Think of the derivative as the slope, and the differential as the actual vertical rise you get by moving along the tangent line by dxdx.

Why It Matters

Differentials make it possible to approximate complicated calculations quickly, such as estimating cube roots or the effect of measurement error on a computed quantity. They also give meaning to the notation inside integrals: in ∫f(x) dx\int f(x)\,dx, the dxdx is literally the differential, representing an infinitesimal width of each rectangle in a Riemann sum. Understanding differentials is essential for techniques like substitution and separable differential equations, where you manipulate dxdx and dydy algebraically.

Common Mistakes

Mistake: Confusing the differential dy with the actual change Δy and treating them as identical.
Correction: The differential dy is a linear approximation of Δy based on the tangent line. They are nearly equal only when dx is small. For large changes in x, dy can differ significantly from Δy because it ignores the curve's curvature.
Mistake: Thinking dx must always equal 1 or some fixed value.
Correction: dx is a free variable — it can be any small quantity you choose. The whole point of the differential formula dy = f'(x) dx is that it works for whatever size dx you plug in, though the approximation improves as dx gets smaller.

Related Terms