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Partial derivatives and chain rule
230869 Partial derivatives and chain rule Find (partial z)/(partial u)( and ) (partial z)/(partial v) using the chain rule. Assume the variables are restricted to domains on which the functions are defined.
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Partial derivatives
Find the indicated partial derivatives.
1.) f(x,y)= sin(y-x); df/dx(3,3)
Using the chain rule,
Therefore, at (x, y) =(3, 3),
2.)z= (x^3 + y^3)/(x^2 + y^2); dz/dx, dz/dy
Using chain rule and quotient rule,
Similar to x, we find the
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Partial Derivatives using Chain Rule
Two multivariable formulas are given and partial derivatives of both are found using the chain rule.
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Apply the chain rule to finding partial derivatives.
The chain rule to is applied to finding partial derivatives.
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Partial Derivatives and Appropriate Chain Rule
9015 Partial Derivatives and Appropriate Chain Rule Find partial deriv's w/r and w/theta using appropriate chain rule for :
w=the square root of (25-5x^2-5y^2), where x=r cos theta, y= r sin theta. w = (25 - 5r^2(cos^2 (theta) + sin^2(theta)))^1/2 or
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Chain Rule : Finding Where f is Differentiable
Using the chain rule, show that if z = y + f(x^2 - y2), where f is differentialbe,
then partial derivatives y(dz/dx) + x(dz/dy) = x
Solution:
The chain rule for a function of two variables is given by,
If z=f(x, y) is differentiable and x
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Chain rule
z); x=x(u, v), y=y(u, v), z=z(u, v)
Thus we have the following partial derivatives:
dp/du=(df/dx)(dx/du)+(df/dy)(dy/du)+(df/dz)(dz/du)
dp/dv=(df/dx)(dx/dv)+(df/dy)(dy/dv)+(df/dz)(dz/dv) This shows how to write chain rule formulas giving the partial
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Function, derivatives, error, and stationary points.
Determine the second-order Taylor polynomial for f near (0,-1)
Use first-order partial derivatives to determine the least and greatest possible values.
Use the chain rule to determine, in terms of t, the rate of change of f.
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Using the Chain rule for derivatives in higher dimension
204667 Using the Chain rule for derivatives in higher dimension Let f(u,v,w) = (eu-w, cos(v+u)+sin(u+v+w)) and g(x,y) = (ex, cos(y-x), e-y). Calculate f o g and D(f o g)(0,0). The solution is attached.