Differentiate the function f(t) = yt (a + bt).

Q: What does it mean for a function of two variables, f (x, y),to be differentiable at a point (a, b)?

A: A function differentiable at a point has a derivative that is defined at that point. This is defined…

Q: Find the gradient of the function at the given point. f(x, y) = 5x + 3y2 + 5, (1, 5) Vf(1, 5) =

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Q: 9) Where is the function ht) = (t+ 5) differentiable?

A: Given         h(t)= t+51/3

Q: Differentiate the function. y = V 3x2 + 8x + 1 dy dx

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Q: Find F'(x). F(x) = dt t4

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Q: Find the gradient of the function at the given point. f(x, y) = 5x + 2y2 + 4, (4, 2) Vf(4, 2) =

A: Topic = Functions

Q: Find the maximum rate of change of the function f(x, y) : 4y4 at the point (1,3)

A: We will solve the problem

Q: 19. y = In(e* + xe*) %3D

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Q: Find the gradient of the function at the given point. f(x, y) = 2x + 5y² +6, (3, 1) Vf(3, 1) =

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Q: Find the gradient of the function at the given point. f(x, y) = 3x + 5y2 + 8, (2, 1) Vf(2, 1) =

A: We have given function  fx,y=3x+5y2+8 fx=∂fx,y∂x fx=3∂x∂x+5∂y2∂x+∂8∂x fx=3+0+0 fx=3  .........(i)…

Q: Find the gradient of the function at the given point. f(x, y) = 4x + 2y2 + 10, (2, 3) Vf(2, 3) =

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Q: The total derivative of f(x,y) = e2x +5y is df =

A: Consider the following function: fx,y=e2x+5y Total derivative of function:…

Q: secx 1 Given g(x) : z dt , find g'(x). t2

A: gx = ∫7sexcx1t2dt           = ∫7sexcx t-2 dt        = t-2+1-2+7secx      =t-17secx-1 = secx-1-7-1-1…

Q: Differentiate the function y = e0.4t. dy dt

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Q: Find the derivative of the function at P0 in the direction of u. f(x, y) = 3x 2 + 4y, P 0(8, 7),…

A: Directional derivative: - The rate at which any function changes at any given location in a fixed…

Q: Find the gradient for the function f(x, y) = x³ – 2xry + 3y².

A: Given : fx,y =x3-2xy+3y2

Q: Differentiate the function. y = 7x dy %3D

A: It is derivative problem.

Q: f(x, y) = x² – y²

A: f(x,y) = x2 - y2 Let us firstly find the first order partial derivatives of the given function.…

Q: Find the domain and range of the function f(x, y) = exy

A: Given information: The function fx,y=exy

Q: Q3. Find the coordinates of the point(s) of the function y=3x – 5x +2, where the tangent has a…

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Q: Find the linearization of the function f(x, y) = /36 – læ² – 2y² at the point (3, -1). L(x, y) = Use…

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Q: For function f(x, y) = 5- | 2 E find the maximum value of the directional derivative at (3, 2).

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Q: Solve for the gradient of the given function. f(x, y) = e" sin y + e cos x

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Q: The antiderivative of the function f(x)= is

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Q: Let f(x, y) = xe*Y. Find the maximum rate of change of f at the point (2, 1). хе

A: We will find the maximum rate of change.

Q: 3 ) Express f(t) in terms of the unit step function u(t).

A:

Q: Find the linear approximation of the function below at the indicated point. f(x, y) = V 17 – x2 –…

A: Givenf(x,y)=17-x2-4y2  Point (2,1)

Q: Find the gradient of the function at the given point. f(x, y) = 5x + 3y2 + 5, (4, 1) Vf(4, 1) =

A: f(x,y)=5x+3y2+5 fx(x,y,z)=5fy(x,y,z)=3(2y)=6y∇f =fx(x,y,z) i +fy(x,y,z) j

Q: Find the linear approximation to the function f(x, y) = sin y --- |T at x, = (1, -2) 2 and use it to…

A: The solution is given as

Q: Find the derivative of the function. -1 y = (3t – 1)(6t – 5) dy dt II

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Q: ) Find the local linear approximation of the function f(x) = V1+x at xo = 3, and use it to…

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Q: Find the gradient of the function at the given point. f(x, y) = 2x + 4y2 + 1, (1, 5) Vf(1, 5) =

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Q: Solve for the gradient of the given function. f(x, y) = e¤ sin y + el cos x

A: Given,  f(x,y)=ex sin y+ey cos x

Q: Find the linearization of the function f(x, y) = V61 – 3x2 - 1y? at the point (4, 2). %3D L(x, y) =

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Q: Let f (z) = (11z+ 10) (i – z). What is the real part of the derivative f' (1 + i)?

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Q: Find R of the function Y = f(x) = vx %3D

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Q: F(x) = f Cost dt 4

A: We need to evaluate F(x)=∫π4x2costdt. So,        F(x)=∫π4x2costdt      =sintπ4x2      =sinx2-12

Q: Find d2y/dx2 by implicit differentiation: a. x3 + y3 = 1 b. y2 = 1 -2/x.

A: a. Given,

Q: Find the maximum rate of change of the function f(x, y) 3y5 x at the point (3,4)

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Q: Find h'(x) if rsin 2x t2 cost dt h(x) = J3 %3D

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Q: Differentiate.

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Q: Differentiate. f(x) = X + r SX +t

A: Given: fx=qx+rsx+t As we know that, The quotient rule: fg'=f'·g-f·g'g2 The Power law: ddxxn=nxn-1

Q: S:R R, f(x) =a", (a > 0, + 1)

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Q: Find the gradient of the function at the given point. f(x, y) = 2x + 4y² + 4, (3, 4) Vf(3, 4) =

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Q: Let f (z) = (7z + 9) (i – z). What is the real part of the derivative f' (1+ i)?

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Q: · y = Vx - x, (1, 0)

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Q: Differentiate the function. y=[(sx +2x+3) (x-3)]² dy dx II

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Q: Find the direction at which the function f (x, у, 2) %3 хIn(е' — хг) increases fastest at (-3,0,0)

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Q: Find the third derivative of f (z) = T2x +I. fll! (z) =

A: Explanation of the answer is as follows