. Differentiate the function using product rule. f(x) = e*(e* + sin x + x²)

Q: Use the definition of the derivative, f'(x) = lim z+h)-(2) to findf' (x) where f(x) = x³ +1. h→0 O…

A: fx=x3+1

Q: If the product function h(x) = ƒ(x) . g(x) is continuous at x = 0, must ƒ(x) and g(x) be continuous…

A:

Q: Determine the derivative of the following function: f(x) = 3ecos(3x) cos(3 x) O 9e cos(3 x) O…

A: f(x) =3ecos3x differentiating function with respect to x

Q: 4. Consider the function f(x) = Vx – I (a) Use the limit definition of derivative to find f'(x)

A:

Q: 4. Suppose fand f'are differentiable functions of x. Given f' (4) = 0, f" (-), and g(x) = (). log…

A: As per the guidelines, we are supposed to answer only one question at once. Kindly repost the other…

Q: (a) Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x…

A: Consider the given piecewise function

Q: Differentiate  f and find the domain of f f(x)  =    ln ln ln x

A:

Q: Find the derivative of the following function g2 F(x) = ds a 3+ 5s4 using the appropriate form of…

A: Given that,

Q: 2 Using the definition of derivative, prove the following: (a) f'(x) = x if f(x) = x %3D

A:

Q: . Show that the following function is differentiable everywhere on R but of which the derivative is…

A:

Q: Let f(x) = x ln(1+ x*). For all æ € [-1, 1], it is known that In(1 + x) = (-1)" -xn+1 n +1 n=0 2.…

A: Substitute x3 for x in the given power series of ln1+x and simplify.…

Q: 1. Evaluate the derivative of f(x) = Vx2 – 1, using the following formula. f(x+h)-f(x) a. lim h-0 h…

A:

Q: Let f(x) be defined by f(x) = x − 2 ln(x), for x > 0 Part a: On what intervals (if any) is this…

A: For increasing f'(x)&gt;0

Q: Find the derivatives of the function f for n = 1, 2, 3, and 4. Use the results to write a general…

A: Given: fx=xnsinx

Q: Use L’Hôpital’s Rule to determine the comparative rates of increase of the functions f (x) = xm,…

A:

Q: Consider ey²cosxy – log3xz = e*z implicitly defines z as a function of x and y . Find and using…

A: The given function is: eyz cos(xy)-log3(xz)=exz

Q: Can you help with this problem step by step?

A: Given,

Q: Find the derivatives of the function f for n = 1, 2, 3, and 4. Use the results to write a general…

A: fx=cosxxn When n=1, fx=cosxx Differentiating with respect to x, we get…

Q: 2. Let f : R → R be a differentiable function. Show that the function f', the derivative function of…

A:

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: if the function f below is continuous at x = 1: f (x) = {1-√1-x, x ≤ 1 ln x, x ≥ 1}

A:

Q: Use L’Hôpital’s Rule to determine the comparative rates of increase of the functions f (x) = xm,…

A: given,         limx→∞x3e2x

Q: [8] Use the definition f(a +h) – f(a) f'(a) = lim h-0 h To find the derivative of the given function…

A: Given: So,          f'(2)=limh→0f(2+h)-f(2)h       =limh→ 012+h-12h       =limh→ 0-12h+2       Plug…

Q: For x > 0, find f'(x) where f(x) = x*¯+3) via logarithmic differentiation. Show all ur work.

A: Topic:- application of derivatives

Q: Find the value of K that make the functionf continuous at x=2 パ-4 マー3x+2 sin(rx) + kx + k if x > 2…

A:

Q: Determine the domain of the function f(x) = In(ln r) + x

A: f(x) = ln(ln x) + x f(x) is not defined if ln x &lt;= 0 [as then ln(ln x) is not defined] ln x &lt;=…

Q: Find all the relative and absolute extrema of the given function on the given domain f(x) = x^2 +…

A:

Q: Determine whether the following statements are true or false:  1. ln(x')=[ln x]'  2. any continuous…

A: The given statement is,

Q: Use the chain rule to find the derivatives of f(x) and g(x) as well as f'(0) and g'(0): f(x) = (2x³…

A: According tp our guidelines we can solve only one question so we are solving 2nd question for you.…

Q: Let f (x) =e*. sin(x²). what is the correct result of the derivative of f with respect to x?…

A: This question is related to differentiation, We will solve it using given information.

Q: Show that the function below is continuous at x 0 (cos x-1 f(x) D if x<0 5x, if x20

A:

Q: Determine the set of values of the continuous function defined by the formula: T COSX b) g(x)= sin 6

A:

Q: f (x) = -1 -1+ x + 4x² cos (:) når x # 0 når x = 0.

A: Consider the given function:

Q: Find all the antiderivatives of the following function. Check your work by taking derivatives. g(x)…

A: Given function is gx=11x10. Antiderivative is opposite of derivative. Antiderivative means…

Q: Use the Power Rule to compute the derivative. (Use symbolic notation and fractions where needed.)…

A:

Q: Find the derivatives of the function f for n = 1, 2, 3, and 4. f(x) = x² sin x n = 1 n = 2 n = 3 n =…

A:

Q: Prove that is a function f is harmonic =0, then - dx -Ldy =0.

A: Given function f is harmonic such that ∂2f∂x2+∂2f∂y2=0

Q: d) Hence form a conjecture for the derivative of functions in the form y = (ax + b)². e) Prove your…

A: As you have mentioned I am answering Question (d) and Question (e)

Q: 4 f(x) = tan(x) cos² (x) 1.

A: Rewrite the given function f(x) using the trigonometric ratios. fx=4cos2 x+1tan x1cos x=sec x and…

Q: Find derivative of the following function by using formula f(x)= limit _f(a) - f(x) а -х a→X (3х-1)…

A: To find:

Q: 2. Find the derivatives of the following functions. 4 a. f(x) = 3x³ x2 b. f(x) = (4 – x²)" C. f(x) =…

A:

Q: 2. Using the definition of derivative, prove the following: 1 (a) f^(x)= if f(x) = Vx %3D

A: fx=x3

Q: Suppose F(x) = e-" dt. Find a formula for the function F(x³) expressed using an integral. F(x³) = /

A: Given that: F(x)=∫17xe-t2dt

Q: To compute the derivative of 2cosx with the chain rule by writing this function as a composition…

A: Consider the two functions f(u) and g(x). Find the composite function f(g(x)) and thus the outer…

Q: a) Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x…

A:

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

A:

Q: Consider the function g defined by 1 g(x, y) = cos (7x/) - CoS log3(x – y)

A: Given problem:-

Q: Find all the antiderivatives of the following function. Check your work by taking derivatives. g(x)…

A: To find antiderivative of function gx=-4cosx-x. Solution: Antiderivative means integration. We have…

Q: Determine the set of values of the continuous function defined by the formula: a) f(x)=cos sin

A: Given that fx=cosπ3sinxgx=π3sinxsinx is continuous in x∈-∞,∞gx is also continuous in  x∈-∞,∞.Range…

Q: Define h(x) = x° sin(1/x) for x # 0 and h(0) = 0. Prove that h is differentiable at x = 0, and find…

A: