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What is the Difference Between Partial and Total Derivative?

Answer #1

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Video - Difference Between Partial and Total Derivative

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The difference between a partial derivative \(\frac{\partial f(x,y)}{\partial x}\) to \(x\) and a total derivative \(\frac{\text{d}f(x,y)}{\text{d}x}\) to \(x\) is that in the partial derivative it is assumed that \(y\) is independent of \(x\).

  • In the partial derivative \(\frac{\partial f(x,y)}{\partial x}\) \(y\) is kept constant.

  • The total derivative \(\frac{\text{d}f(x,y(x))}{\text{d}x}\) does NOT assume that \(y\) is constant. The change of \(x\) also affects \(y\).

Example

For example, consider a function \(f(x,y) = 3x^2 ~+~ 2y \) that depends on \(x\) and \(y\). A partial derivative of this function with respect to \( x \) is:

Partial derivative of a function
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The derivative of \(2y\) with respect to \(x\) is zero because \(y\) is independent of \(x\). A total derivative of the function, on the other hand, is:

Total derivative of a function
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Here the second term is generally not zero, because \(y(x) \) can be a function of \( x \). If \(y\) is independent of \(x\), then the total derivative coincides with the partial derivative. Thus, a partial derivative is a special case of the total derivative.