Let y = 2 ln x - 1/2x then the first derivative is....., the first order condition is..... and the value of x satisfying the FOC is...

Q: 1. Find an equation of the tangent line to the graph of the function at the given point. f\xl=In…

A: Solved below

Q: Find the first derivative of each function. a) f(x)=ln (e^x+e^-x)

A: “Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: 5. Use logarithmic differentiation to find the derivative of these functions. (a) y = /Te* (x² +…

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Q: The derivative of the function y = log2(In(x)) is: 1 1 y': x In 2 In x y' = x In 3 ln x the above…

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Q: Find the derivative of the function. y = sin−1(3x + 1)

A: Given data:     The given function is: y=sin-1(3x+1).   The given function can be expressed as,…

Q: 4) fG) = (2x+ 1)9 %3D

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Q: ne following second-order differential equa

A: Introduction: A second-order differential equation has a form d2ydx2+p(x)dydx+q(x)y=r(x), where…

Q: 3. Find the derivative. sh (a)l= - पक) (a) f(x) = x2 cos x +x tan r (b) h(x) = e" (x² + 4x) %3D t2…

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Q: The derivative of the function f(x) =x ln x is: a. f'(x) = In x +1 b. f'(x) = ln x +. c. f"(x)= In x…

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Q: (b) Sketch the graph of f (x) = ln(4 – x²). Use the first and second derivatives to identify where…

A: Consider the function  fx=ln4−x2

Q: 7) Find the average rate of change of the function K(x) = cos (x--) from x = 0 to x = 6

A: First of all find values of function at end points then put in formula of average rate of change.

Q: (a) Find the derivative. y In(4x – 5) – 3 In(x) dy dx (b) Find the derivative. 4x - 5 y = In x3 dy…

A: a) First we shall find the derivative of y=ln4x-5-3lnx Now,…

Q: 1. (a) Show that the following function is continuous at x = 1 for all values of p. spx + 1, if x >1…

A: A function f is said to be continuous at a point x=a if limx→af(x)=f(a).  The left hand derivative…

Q: What is the approximate value of the first derivative of the y = sinx function at x =pi/4 when you…

A: The first derivative of the function can be obtained using the forward and backward difference…

Q: 4. Find the equation of the tangent line and the normal line to the graph of the given function at…

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Q: 1. • Let a be an unknown constant and 1 f(x) = VT + +a. Find the derivative of f(r) with respect to…

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Q: Verify that each given function is a solution of the given differential equation. Throughout these…

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Q: Use one or more logarithm identities to show that In (2) In(x) for any > Use one or more logarithm…

A: we have to do as per the question.

Q: 1. Given: f(x) = Vx – 1. a. Determine f (x) using the definition of derivative. b. Determine the…

A: Given,f(x)=x-1differentiating both the sides w.r.t. xf'x=12x-11f'x=12x-11

Q: 1) The derivative of f (x) = e2x² cos x is: 2) Simplifying the expression e n(4x-3) + eln 2x In(e3x)…

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Q: Use the Chain Rule to find the derivative. y = cos (e40) (Express numbers in exact form. Use…

A: Given.

Q: |(x+ 1) |(x – 1) Find the derivative of k(x) = ln A. k'(x) = x2-1 -1 В. К(х) — (x²+1)

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Q: n In a certain tropical forest, litter (mainly dead vegetation such as leaves and vines) forms on…

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Q: 2.6 The Chain Rule: If y= f(u); u= g(x), and the derivatives and both exist then the du dx composite…

A: Given y=f(u) and u=g(x) and dydu and dudx both exist.

Q: Here are three functions involving exponential functions: s(x) i) s(x) = 글 (e" -e") i) c(x) = } (e*…

A: Introduction: The trigonometric functions are also differentiable over their principle domain. The…

Q: Let f be the function defined for x > 0, with f (e) = 2 and f', the first derivative of f , given by…

A: Given, f be a function defined for x>0 and the derivative of f is f'x=x2lnx Point e, 2

Q: 1. Compute the first, second, and third derivative of a. f(x) = x2 In(x+1) %3D b. h(x) = x2 sin x…

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Q: Here are three functions involving exponential functions: i) s(x) = = (e" – e*) $(x) iii) (x) = c(x)…

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Q: 2 Compute the derivative f'(x) of the logistic sigmoi 1 f (x) : 1+ exp(-x)

A: We have to find

Q: a) Find implicitly, if y°cosy +x = 2xy b) Define differentials also find derivative by using…

A: (2)(a) We have the given function as y2cosy+x2=2xy On using implicit differentiation, we get the…

Q: Find the linearizationsof the following functions at x = 0. ln (1 + x)

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Q: 3 x° In x x° with respect to X - Find the derivative of y a) 6x - a2 b) None of these c) p6 ln x x2…

A: Given query is to find the derivative of the expression.

Q: 8. Find the derivative of the f(x) = ntan x at x = " А. п B. 2 In(1) C. π In (π) D. 2 n In() E. 2n…

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Q: 5- When the first derivative of the y = sinx function at x =pi/4 is calculated with the forward…

A: We will use basics of numerical methods here

Q: 7. Find the second derivative given g(x) = sin x² A. 2(cos x? – x sin x?) B. 2(cos x? – 2x² sin x²)…

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Q: 1. Compute the first, second, and third derivative of a. f(x) = x² In(x+1) b. h(x) = x² sin x

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Q: Find the derivative of y. x2 =log6 x+1 O A) Sx+1 x2 O B) ) In 6 x2 2x+1 1 Sx+1 In 6 x2 D) None of…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 14))) What is the approximate value of у= sinx when you calculate the first derivative of the…

A: We will use basic knowledge of numerical methods to answer this question.

Q: 3) Show that for any constants M, k, and a, the function k(t – a) y(t) = -M1+ tanh 2 satisfies the…

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Q: Find the derivative of the logarithmic function. y = In 6 + e 6- e*. Step 1 The given function y =…

A: We will derivate the function step wise as asked in the question.

Q: 1. Find the derivative of the following (a) f(x) = log-(r* +2) + 7** (b) k(r) = In(x) + 5e¯ (c) j(z)…

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Q: 4. Find the second-order derivative of the following function: f(z) = log, cos z + In %3D

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Q: Derivative OF the FollOwing 1.y sin x)* 2. y-Tex te* 3. d dx ( scextanx t in csccxttanx)

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Q: 1.find the derivative of In[In(Inx)] 2. find the derivative of e(2x+Inx

A: Using chain rule of differentiation.

Q: x² +2x+1 6.) f(x) = 3

A: Given :f(x)=x2+2x+13

Q: E2-3

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Q: e) The function f(x) = log(x)log(x) with x %3D Jse fzero to find a zero of its derivative. What is…

A: The given function is f(x)=log(x)log(x); with x≥1

Q: 1.) f(x) = (2x + 1)(3x – 2)

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

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Q: Find the first derivative or f’(x) of the following functions using the techniques of integration:

A: Given,            fx=40015-x23x-2To find : The first derivative or f’(x) of the given function.