What is the interpretations of a partial derivative with respect to x? With respect to y?
The partial derivative with respect to x represents the rate of change of the function with respect to changes in the x-coordinate while holding all other variables constant. Similarly, the partial derivative with respect to y represents the rate of change of the function with respect to changes in the y-coordinate while holding all other variables constant.
Geometrically, the partial derivative with respect to x can be interpreted as the slope of the tangent line to the curve when x is varied while y is held constant. Similarly, the partial derivative with respect to y can be interpreted as the slope of the tangent line to the curve when y is varied while x is held constant.
The interpretation of a partial derivative with respect to x is the rate of change of a function with respect to the x-variable, while keeping all other variables constant. It measures how the function changes as we vary the x-value, while holding the other variables fixed.
Similarly, the interpretation of a partial derivative with respect to y is the rate of change of a function with respect to the y-variable, while keeping all other variables constant. It measures how the function changes as we vary the y-value, while holding the other variables fixed.
A partial derivative measures how a function changes with respect to a specific variable while holding all other variables constant.
To interpret a partial derivative with respect to x, you can think of it as determining the rate of change of the function in the x-direction while keeping the other variables constant. It tells you how the function values rise or fall as you move along the x-axis.
To find the partial derivative with respect to x, you can differentiate the function with respect to x while treating all the other variables as constants. You can do this by applying the rules of differentiation, such as the power rule or the chain rule if necessary.
Similarly, to interpret a partial derivative with respect to y, you are interested in the rate of change of the function in the y-direction. It tells you how the function values rise or fall as you move along the y-axis.
To find the partial derivative with respect to y, differentiate the function with respect to y while treating all the other variables as constants.
In summary, the interpretations of partial derivatives with respect to x and y involve understanding how a function changes with respect to each specific variable while keeping all other variables constant. Differentiating the function with respect to the desired variable will give you the corresponding partial derivative.