Let f be a C' function on the line, and let g(x) = S f(xy)y² dy. Prove that g is a C' function and establish a formula for g (x) in terms of f.

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Q: hi, i've added the problem as a picture please take a look. thanks in advance!

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A: solution :-

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A: Given:        x2+2xy=y3

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A: linearlization of f(x) at x=a is  L(x) = f(a) + (x-a)f'a)

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A: The differential of y is the derivative of the function times the differential of x,

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Q: Find the linear approximation to f(x) = cos(x) near a =- [Where L(x) = f(a) + f'(a)(x – a) ] 3

A: Solution: The objective is to find the linear approximation

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Q: Let y = f(x), where f(x) = 7x5/6 + 5x²/3. Find the differential of the function. dy = X

A: The objective is to find the differential function.

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A: Plug the given expressions of x,y, and z. Then differentiate directly. And plug t=1

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Q: suppose that x and y are both differentiable functions of t and are related by the given equation.…

A: Suppose that x and y are both differentiable functions of t and are related by the given equation…

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A: As per guidelines we can solve only one question at a time so if you need more answers kindly repost…

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A: LINEAR APPROXIMATION

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Q: Let W=x*yz.compute the differential dw.

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