In Exercises 35-40, find the partial derivative of the function with respect to each variable.

In Exercises 35-40, find the partial derivative of the...

Key Concepts

Multivariable Functions

Multivariable functions are functions that depend on more than one independent variable. They are common in many areas of science and engineering, and understanding their behavior requires using techniques like partial differentiation which isolate the rate of change with respect to a single variable.

Partial Derivative

A partial derivative is the derivative of a multivariable function with respect to one of its variables, holding the other variables constant. This concept is fundamental in analyzing how a function changes along one specific dimension while ignoring variations in the other dimensions.

Product Rule

The product rule is used in differentiation when a function is the product of two or more functions. It states that the derivative of a product is the derivative of the first function times the second function, plus the first function times the derivative of the second. This rule is crucial when computing partial derivatives for functions that include multiplicative factors.

Treatment of Constants in Partial Differentiation

When taking a partial derivative, all variables other than the one being differentiated with respect to are treated as constants. This simplification allows for focusing on the effect of changes in the one variable of interest while keeping other factors fixed.

Quotient Rule

The quotient rule is applied when differentiating a function that is expressed as one function divided by another. It provides a systematic way to handle the differentiation of ratios, which is important when the variable of interest appears in the denominator of a fraction.

Transcript

00:01 All right, so we're given way the function w equals ev plus v delta new squared or wuging g.

00:09 Okay.

00:10 So we're going to get the partial derivative of this function of w with respect to each of the variable on the right hand side.

00:18 So we have variables p, v, delta, new, and g.

00:25 Okay, so let's start with the partial derivative of w with respect to.

00:32 P.

00:33 Okay.

00:34 So for the first term, okay, we have the partial derivative of p -v, okay, with respect to p.

00:42 So that's, we'll treat the other variables as constant, and we'll just get the derivative of each term with respect to p.

00:51 So we have v, a partial derivative of p with respect to p.

00:58 Okay.

00:59 Plus the partial derivative of p with respect to p okay so notice that there uh there is no variable p in this term so put it this whole term as a constant and the derivative of a constant is just zero okay so we're not we're not going to write it here so we have b partial derivative p with respect to p so that's just equal to one therefore we have v okay next the partial derivative of w with respect to v okay so for the first term we got p partial derivative of v with respect to v plus for the second term we'll treat the other variables as constant so we have delta new squared over 2g and then partial derivative of v with respect to v okay so these terms are just equal to one and also this one therefore we have p plus delta b squared over 2g okay next we have the partial derivative w with respect to delta okay now for the first term which is t b we don't have a variable delta so we'll cheat t b as a constant and differentiating that we get zero okay so let's not write that anymore and then we have the second term so we'll treat the other variables as constant so we have v new squared over 2g then partial derivative of delta with respect to delta okay and we know that this is just equal to one hence we have v new squared over 2g okay next we have partial derivative with respect to new the small are v here...

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