Find (∂f)/(∂x) and (∂f)/(∂y) for each of the following functions: a f(x, y)=x^2+y^2 b f(x, y)=si

Find (∂f)/(∂x) and (∂f)/(∂y) for each of the following...

Key Concepts

Constant Multiple Rule

The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. In partial differentiation, this rule enables the straightforward treatment of constants, ensuring that they are appropriately factored out during the differentiation of each variable-dependent term.

Partial Derivative

A partial derivative measures the rate at which a function of multiple variables changes as one particular variable changes, while all other variables are held constant. This concept is fundamental in multivariable calculus and is used to analyze how a function behaves in different coordinate directions.

Power Rule

The power rule is a basic rule in differentiation used to compute the derivative of a function that is raised to a constant exponent. When applied in partial differentiation, each variable is treated independently, and the power rule allows us to differentiate each term of the form x^n or y^n by bringing down the exponent and reducing it by one.

Sum Rule

The sum rule states that the derivative of a sum of functions is equal to the sum of the derivatives of those functions. This rule simplifies the process of differentiating complex expressions by enabling the differentiation of each term separately before recombining them.

Product Rule

The product rule is used when differentiating an expression that involves the product of two functions. In the context of partial differentiation, if a function is expressed as a product of variable-dependent factors, the product rule provides a method to differentiate by taking the derivative of one factor while keeping the other constant and vice versa.

Trigonometric Differentiation

Trigonometric differentiation involves applying specific derivative rules to functions like sine and cosine. These rules are essential when handling differentiable trigonometric functions in multivariable contexts, ensuring that the derivatives of functions such as sin(x) or cos(y) are computed correctly.

Transcript

0:00 Hi there.

00:01 In this problem, we're asked to find the partial derivatives with respect to x and y of this function.

00:06 So let's begin with the partial respect to x.

00:10 Now before we even do this, why don't we rewrite the original function just a little bit so that it looks like what it means.

00:17 The way we write the 2 is always a little confusing in my opinion.

00:22 So remember what this means is sine of x minus 3y and that whole result is squared.

00:29 So that's what the original function actually says.

00:32 When we write it like this, it looks a little bit more like the chain rule that we're going to need.

00:38 When we take the derivative, we start as we always would from the outside in.

00:43 In other words, this two comes out in front where let's treat this whole thing like a power rule.

00:47 Two to a bunch of stuff is two times that stuff.

00:52 X minus three line there.

00:54 Okay.

00:55 All to the first power, we drop the power by one.

00:57 And now we need to multiply by the derivative of all this inner stuff.

01:05 Now when we say derivative, in this case we mean the partial derivative with respect to x of all this stuff in the middle.

01:17 So we'll just copy everything here.

01:20 The power is one here so we can just get rid of those brackets.

01:26 Okay, now what is the partial derivative of with respect to x of sine of x minus 3y? notice we have another chain rule.

01:33 We have this x minus 3y inside of a sign.

01:38 So the derivative of sine, first of all, is cosine.

01:44 We always leave the inner function intact.

01:48 And now the second rule, the chain rule is kicking in.

01:51 We need to multiply by the derivative of this stuff.

01:56 So let's keep writing partial derivative with respect to x.

01:59 We always mean partial with respect to x in this case when we say derivative of x minus 3y.

02:06 Okay, so we're almost done.

02:07 We just had to unravel this using the chain rule twice...

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