The linear approximation L(x) of a differentiable function f(x) near the point x = a is L(x) = f'(a) + f (a)(x – a).

Q: Determine the instantaneous rate of change of the function πt ³ √x sin 2 f(x) = LV x3 1 + √t at x=2.…

A: Leibnitz integral rule.

Q: If f(x, y) = y° cos(x) – x³ In(y), find the second partial derivative ду дх .A ² f ду дх 3x2 -2y…

A: Calculate partial derivatives

Q: equilibrium

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Q: If (x, y) moves from (x0, y0) to a point (x0 + dx, y0 + dy) nearby, how can you estimate the…

A: The resulting change in function f (x, y) is given as:

Q: Find the differential of the function: y = x In x

A: Solution:Giveny=x ln xTo find the diffferentiationformula:(uv)'=uv'+vu'ddxxn=nxn-1ddxln x=1x

Q: Find partial derivative af dx for the function f(x,y) given below exp 2x. siny .(at the point (In2,…

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Q: Find the partial derivative of f with respect to x of the function f(x,y) =xy(e^x3)+y

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Q: 26) Find the partial derivative of of f= y cos x+ 2x sin z + 5y with respect to y.

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Q: Find the maximum rate of change of the function f(x,y) = sin (xy) at the point (4,0) and the…

A: The given function is f(x,y)=sin (xy) and the point is (4,0)=(a, b). to find : Maximum rate of…

Q: If f(x, y) = y² cos(x) – a° In(y), find the second partial derivative

A: Partial derivative problem

Q: a?f (or and y. that is, f(x,y) = e² with respect to ayax axay Calculate the second-order partial…

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Q: (a) Show that the derivative solution of this function y = x** is dy = x(**+x=1) [1+ x ln x (In x +…

A: The given function y=xxx Note:    Multiple questions are given, we are answered with only one…

Q: find the partial derivative of the function with respectto each variable. ƒ(x, y) = 1/2 ln (x2 + y2)…

A: The given function is fx,y=12lnx2+y2+tan-1yx.

Q: COs X Compute the mixed second order partial derivative of z =

A: The mixed second order partial derivatie of z=cos x y5  is given by ∂2z∂x∂y=5sin xy6

Q: Find the linear approximation of f(x)=ln(x) x=1 and use it to estimate ln(1.1). L(x)= ln(1.1)=

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Q: Find partial derivative of dx for the function f(x.y) given below exp 2x. siny .(at the point (In2,…

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Q: Let f:[-n, 7] –→ R, where f(x) = 2 sin (2x) – 1. TC Calculate the average rate of change of f…

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Q: Find the linear approximation of f(x) = ln x at x = 1 and use it to estimate ln(1.33).

A: The linear approximation of f(x) at x=a is given by L(x)=f(a)+f'(a)(x−a) Given, f(x)=ln(x) at x=1.…

Q: Find the first and second order partial derivatives for z x2 sin (3x + 2y).

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Q: Compute the partial derivative fxxyz for f(x, y, z) = cos(cos(x3 + z2 )) + x2 ln(y2 + z2 ).

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Q: let f(x,y)= z-sin(xy),x= t, y= lnt, z=e^t-1. find derivative of f directly and by using chain rule…

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

Q: If f(x, y) = y cos(x) – a° In(y), find the second partial derivative ду дх .A 3x2 = -2y sin(x) ду дх…

A: Option (a) is correct.

Q: Find the first order partial derivative with respect to y S(x, y) = a"y² + 3ze"

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Q: Find the partial derivative fx ( x , y ). f ( x , y ) = x ln ( 3x + 4y )

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Q: Find the Partial derivative of f with respect to y for * 2. y² f (x, y) = x + y

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Q: Find the partial derivative Əz Əx for the function z = 8y²√x.

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Q: Q#2 Find the relative extrema using both First and Second Partial Derivative test i) f(x) = x* –…

A: Given: A function fx=x4-12x3 To find relative extrema first find critical points and then substitute…

Q: = Determine the equation of the tangent line to the graph of f(x) = I x+ln x cosh (√2t - 1) dt at x…

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Q: Find the partial derivative of f(x,y) with respect to y at the point (3,2) f(x,y) = ln(x^2y-xy^2)

A: A partial derivative of a function of several variables is its derivative with respect to one of…

Q: Find the partial derivative. = (a² + 4x – 2y)° - dz II

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Q: Find the first-order partial derivative with respect to x of f(x,y) = e5x siny.

A: To find the first-order partial derivative with respect to x, given function, fx,y=e5x sin y

Q: From the partial differential eqiuation by eliminating the arbitrary function z=(x2+y2+z2)

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Q: Use the second partial derivative test to find the relative extrema of f(x,y) = -x³ + 4xy 2y2 + 1.

A: The given function is  f(x, y)=-x3+4xy-2y2+1

Q: If f(x, y) = y² cos(x) – x³ In(y), find the second partial derivative ду дх

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Q: Find the first and second order partial derivatives for z = x² sin (3x + 2y).

A: As we know that; The product rule: f·g'=f'·g+f·g' The chain rule: fgx'=f'gx·g'x And The first order…

Q: Find partial derivative af for the function f(x,y) given below exp 2x. siny .(at the point (In2, 0

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Q: Find the linear approximation of f(x)=ln x at x=1 and use it to estimate ln(1.24). L(x)=?…

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Q: Find the partial derivative with respect to x of e^y-2*ln(x)-x^3y^2.

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Q: Find both first partial derivative h(x, y) = e-(xª + y^) hx(x, y) = hy(x, y)

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Q: Find partial derivative of: z=f(x,y)= e^x^2+y^2  +   cos(x+y^2

A: For this question of partial derivative we need to find out first derivative with respect to x and y…

Q: Calculate the second-order partial derivative of f(x.v) =sin(x³v?) with respect to x and y, that is,…

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Q: (4)The instantaneous rate of change of f(r) = e- at a = 1 is . .. (5) The derivative of f(x) = sec2…

A: (4) We have So we get the instantaneous rate of change of the function at  as .

Q: (or axay дудх Calculate the second-order partial derivative of f(x.v) = xy with respect to x and y,…

A: Given  f(x,y)=xy

Q: Find the mixed partial derivative at the point (2, 3) for h(x.y) = x sin(y) + y sin(x) + xy

A: The given function is, Differentiate the above expression with respect x partially.

Q: find the partial derivative of the function with respectto each variable. h(x, y, z) = sin (2πx + y…

A: We have to find the partial derivative of the function with respect to each variable   The given…

Q: Explain of the method of proving the derivative (dy/dx) of y= U(x)"(») where U(x)and V(x) are…

A: The function is, y=U(x)V(x)

Q: Calculate the derivative of the function g(x)=x∫√x(t2−t)dt at x=1.

A: using the fundamental theorem of calculus we find g'(x)