Use the chain rule to find dzdt z x2 y2 xy x sint y 8et


What is the chain rule?

The chain rule is a mathematical rule that allows you to calculate the derivative of a composite function. A composite function is a function that is made up of two or more functions. For example, the function f(x)=g(h(x)) is a composite function. The chain rule says that the derivative of a composite function can be calculated by taking the derivative of the outermost function and multiplying it by the derivative of the innermost function. In our example, we would have: f'(x)=g'(h(x))*h'(x)

How do you use the chain rule to find dzdt z x2 y2 xy x sint y 8et ?


To find the derivative of z with respect to t, we can use the chain rule. Since z is a function of x and y, and x and y are both functions of t, we can write:

dzdt = dzdx * dxdt + dzdy * dydt

To find dzdx and dzdy, we can take the partial derivatives of z with respect to x and y. For this function, we get:

dzdx = 2x * y2 + xy * (cos t) + 8e * (sin t)
dzdy = 2y * x2 + xy * (-sin t) + 8e * (cos t)

plugging these back into our original equation, we get:
dzdt = (2x * y2 + xy * cos(t) + 8e * sin(t))* (x’)+(2yx2+xy(-sin(t))+8ecos(t)) (y’)

Examples of the chain rule in action

Calculus is all about finding rates of change. The chain rule is a tool that allows us to find the derivative of a function that is the composition of two or more functions. In other words, it allows us to find the derivative of a function that is built up from smaller functions.

For example, consider the function f(x) = x^2 + 3x + 5. We can see that this function is made up of two simpler functions: g(x) = x^2 and h(x) = 3x + 5. Using the chain rule, we can find the derivative of this composite function:

f'(x) = g'(x) * h'(x)
= 2x * (3x + 5)’
= 2x * 3
= 6x


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