طرح‌ها

ماشین‌حساب مشتق جزئی

این ماشین‌حساب از چندجمله‌ای f(x, y) نسبت به x و y مشتق می‌گیرد، هر دو مشتق جزئی را در یک نقطه حساب می‌کند و مشتق جزئی مختلط ∂²f/∂x∂y را پیدا می‌کند — و بررسی می‌کند که هر دو ترتیب مشتق‌گیری یکسان باشند.

نتیجه

مقادیر را وارد کنید و برای دیدن نتیجه «محاسبه» را بزنید.

فرمول

∂∂x(xmyn)=m xm−1yn\frac{\partial}{\partial x}\left(x^m y^n\right) = m\,x^{m-1} y^n

To differentiate with respect to x, treat y as a constant (and vice versa).

∂2f∂x∂y=∂2f∂y∂x\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}

For polynomials (and other smooth functions) the order of the mixed partials doesn’t matter.

نحوه استفاده از این ماشین‌حساب

  1. Type f(x, y) as a polynomial in x and y, such as x^2y + 3xy^2 (xy means x × y).
  2. Choose the point (x, y) where you want the values.
  3. Press Calculate to see ∂f/∂x, ∂f/∂y, their values at the point and the mixed partial derivative.

مثال‌های حل‌شده

Partial derivatives of x²y + 3xy²

∂f/∂x: treat y as a constant: 2xy + 3y².

∂f/∂y: treat x as a constant: x² + 6xy.

At (1, 2): ∂f/∂x = 4 + 12 = 16 and ∂f/∂y = 1 + 12 = 13. Mixed partial: 2x + 6y.

A point where one partial is zero

∂f/∂x = 3x² − 2y and ∂f/∂y = −2x + 4y³.

At (2, 1): ∂f/∂x = 12 − 2 = 10 and ∂f/∂y = −4 + 4 = 0.

اشتباه‌های رایج

  • Differentiating the other variable too: in ∂/∂x, y is a constant, so ∂/∂x (3y²) = 0.
  • Dropping a constant factor: ∂/∂x (x²y) = 2xy, not 2x.
  • Substituting the point before differentiating.
  • Mixing up the notation: f_x means ∂f/∂x.

پرسش‌های متداول

What is a partial derivative?

The rate of change of a function of several variables as one variable changes and the others are held fixed. ∂f/∂x is the slope in the x-direction.

How do I find a partial derivative?

Differentiate as usual with respect to one variable, treating every other variable as a constant number.

Which functions can I enter?

Polynomials in x and y, up to total degree 12, such as x^3 - 2xy + y^4. Answers are exact.