Differentiation methods

How to Find Partial Derivatives: Steps and Examples

Partial differentiation is ordinary differentiation with one careful choice: decide which independent variable is changing. Every other independent variable is treated as a constant during that calculation.

Direct answer

Choose one variable; hold the other independent variables constant.

A partial derivative describes change in one independent direction while the remaining independent variables stay fixed.

Choose the derivative task before touching the algebra

The symbols alone do not tell you which derivative to compute. First identify the active variable and whether the remaining variables are independent coordinates or functions of that variable.

Ask first 1

Does the problem request ∂/∂x or ∂/∂y?

What to do

Mark that symbol as active and hold the other independent coordinates fixed.

Ask first 2

Does the equation state or imply y=y(x)?

What to do

Use ordinary or implicit differentiation instead of treating y as an independent constant.

Ask first 3

Does the problem ask for a mixed or second partial?

What to do

Compute the requested first partial, then differentiate that result in the stated second direction.

Worked method: f(x,y)=x²y+y³

For ∂f/∂x, x is active and y is constant. The derivative of x²y is 2xy, while y³ contributes zero, so ∂f/∂x=2xy.

For ∂f/∂y, y is active and x is constant. Now x²y contributes x² and y³ contributes 3y², so ∂f/∂y=x²+3y².

This is different from implicit differentiation: here x and y are independent coordinates. If an equation instead makes y depend on x, differentiating a y-term introduces dy/dx.

1
f(x,y) = x²y + y³
2
∂f/∂x = 2xy + 0 = 2xy
3
∂f/∂y = x² + 3y²

Worked examples

∂∂x[x3 y2]\frac{\partial}{\partial x}\left[x^{3}\,y^{2}\right]
3 x2 y23\,x^{2}\,y^{2}

Treat y² as a constant multiplier.

∂/∂y (sin(xy))
x cos(xy)

The inner derivative with respect to y is x.

∂²/∂y∂x (x²y+y³)
2 x2\,x

For supported smooth polynomials, differentiate the first partial again in the requested direction.

Common mistakes

  • • Differentiating every variable at once instead of choosing the active variable.
  • • Treating a dependent y=y(x) as an independent constant; that is an implicit-differentiation problem.
  • • Assuming mixed partial derivatives always agree without checking smoothness near the point.

Error clinic: held fixed does not mean deleted

Incorrect attempt

∂∂x[x2 y+y3]=2 x+3 y2\frac{\partial}{\partial x}\left[x^{2}\,y + y^{3}\right] = 2\,x + 3\,y^{2}

Why it fails

The work differentiates y even though x is the active variable, and it drops the fixed multiplier y from x²y.

Correction

∂/∂x (x²y+y³) = 2xy

Incorrect attempt

∂/∂y sin(xy) = cos(xy)

Why it fails

The inner derivative of xy with respect to y is x, not 1.

Correction

∂/∂y sin(xy) = x cos(xy)

Now change the active variable yourself

Write “active” and “held” beside each problem before differentiating. The two questions use the same method but preserve different symbols.

∂∂x[x3 y+5 y2]\frac{\partial}{\partial x}\left[x^{3}\,y + 5\,y^{2}\right]

Check before solving

Hold y fixed and decide which term becomes zero.

Reveal the reference answer

Reference answer

3 x2 y3\,x^{2}\,y
∂/∂y sin(xy)

Check before solving

Keep x as the inner derivative with respect to y.

Reveal the reference answer

Reference answer

x cos(xy)

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Partial derivative calculator:x2 y+y3x^{2}\,y + y^{3}Partial derivative calculator:x2 y+y3x^{2}\,y + y^{3}
Open the calculator without a preset: Partial derivative calculator

References

Frequently asked questions

Does holding y constant mean setting y=0?+

No. Its value is fixed during this comparison, so it can remain as a coefficient in terms that still change with x.

How is a partial derivative different from implicit differentiation?+

A partial derivative varies one independent coordinate. Implicit differentiation treats variables such as y as functions of x because an equation links them.

Can I take a partial derivative twice?+

Yes. Differentiate the first partial again in the requested direction, while checking that the needed derivatives exist near the point.

Conclusion

Choose the active variable first, state what is held fixed, and process every term under that condition. That single decision separates partial differentiation from several look-alike tasks.

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