Partial Derivatives Part 2

Main Ideas

A partial derivative measures how a multivariable function changes as one variable changes while the other variables are held constant.

• To find a partial derivative with respect to (x), treat (y) as a constant.
• To find a partial derivative with respect to (y), treat (x) as a constant.
• Geometrically, (\frac{\partial f}{\partial x}) gives the slope in the (x)-direction, while (\frac{\partial f}{\partial y}) gives the slope in the (y)-direction.

Procedure and Examples

Differentiating with Respect to (x)

Treat every (y) as a constant, and differentiate only the parts that depend on (x).

Differentiating with Respect to (y)

Treat every (x) as a constant, and differentiate only the parts that depend on (y).

Example 1

For (f(x,y)=x^2+y^2),

[ \frac{\partial f}{\partial x}=2x, \qquad \frac{\partial f}{\partial y}=2y. ]

Example 2

The second example appears to use

[ f(x,y)=e^{3x}+e^{2y}+7xy. ]

Applying the same rules gives

[ \frac{\partial f}{\partial x}=3e^{3x}+7y, \qquad \frac{\partial f}{\partial y}=2e^{2y}+7x. ]

The subtitles give inconsistent exponents for the (e^{2y}) term, so this expression is inferred from the derivatives described in the lesson.

Speaker/Source

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