Let f(x) = 5 · In(e² · x). Let g =f-1. Find g(10), f'(x) and g'(10)

Q: f(x) = acos(b\x) x2 COS

A:

Q: Find the first derivative of g(x) = - 5 In(x)cos(r"). g'(x) =

A:

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 (y') by Implicit differentiation e(2x+3y) = x² – In(xy³) 1 Add file

A:

Q: 3. Find the derivative of g(z) = V9- using the definition of derivative.

A:

Q: Find y" by implicit differentiation. x² + 4y² = 4

A:

Q: Find the derivative of: 9e 7sin ( 3x*) Use e^x for e". -

A: We use chain rule to find the derivative.

Q: Use the chain rule to find the derivative of the function z = x2y3 − x along the curve given by x =…

A:

Q: use the Chain Rule to find the derivative. y = cos3(e4θ )

A: Given: The function given is  y=cos3(e4θ)   To determine: The derivative of the function y=cos3(e4θ)…

Q: find the derivative of y with respect to x, t, orθ, as appropriate. y = esin t(ln t2 + 1)

A: The variable is t in this equation  So we will differentiate with respect to t   We will use the…

Q: Assume mmh (x) = f(g(x)) such that the first an exist g Second derivatives of both of and What is b"…

A: Assume h(x) = f(g(x)) such that the first and second derivatives of both f and g exist  We have to…

Q: Use the Power Rule to compute the derivative: d. X. dx x=-1

A:

Q: Find the differential d8y of the function y=exsinx.

A:

Q: Find the first derivative of the following functions with respect to x y = In(ln(In x))

A: To find first derivative of the given function.

Q: Find the differential equaiton corresponding th y= a cos(log x) +b sin(log x) where a and b are…

A: Differential equation formation

Q: Use the Chain Rule to find the derivative. y = cos3(e4θ )

A: We have given y=cos3e4θ Then, dydθ=dcos3e4θdθ=3cos2e4θ-sine4θ4e4θ=-12e4θsine4θcos2e4θ

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)=

A:

Q: Compute the derivative of the function y=x2/ln x at x=e.

A:

Q: Using limit definition of derivative, prove d/dx e^x=e^x

A: We using limit definition

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: The derivative of the function 3+ln x is: y = cos(In x)

A: y=3+ln(x)cos(ln(x)Differentiate w.r.t.xApply the Quotient Rule:  fg'=f…

Q: Determine the derivative of g ( x ) = 3 e-4 x*sin(7 x) with respect to x. = (x), 6

A: As per the guidelines, we need to author the 1st question.  We know that product rule is given by…

Q: Find the second derivative 1. y = In(x^3+1) 2. y = Arctan (e^x)

A: Given: y=lnx3+1, y=tan-1ex

Q: find the derivative of y with respect to x, t, or θ, as appropriate. y = x4/4 ln x - x4/16

A:

Q: find the derivative of y with respect to x, t, or θ, as appropriate. y = ln (t3/2) + √t

A:

Q: Find the interval in which function f(x)=sin (log,x)+cos(log,x) is decreasing

A:

Q: 9. Find the derivative of the function using chain rule. f(t) = 3t In t

A: To find the derivative of the function

Q: Find the derivative of the function y=cos(3 In x).

A: Given y=cos(3lnx)

Q: (a) By using the chain rule and the derivative = (xl) if x>0 (-1, if x< O' dx find - (Ix – x|)DvxE…

A: Given : ddx(x)=1 ,     if x&gt;0-1 , if x&lt;0 To find : ddx(x3-x) , where x∈(-1,1) and x≠0

Q: y = cof'/ x2-1 x²+

A:

Q: Find the derivative of f(x) and solve for x sinx a. f(x) X = 6. b. f(x) e* sinx X = 0 COSX c. f(x) X…

A: Given: (1) f(x) = sin x / x  at x = π/6  (2).  f(x) = ex sinx at  x= 0 (3).  f(x) =  cos x / ex   at…

Q: Let y = In(x'e2x ). Find dy/dx.

A:

Q: If u, v, w are nonzero differentiable functions, then the first derivative of uv / w is

A:

Q: Find y″ by implicit differentiation. x2 + xy + y2 = 8

A: Given:    Implicit differentiation :     x2+xy+y2=8   differentiating on both sides we get…

Q: (a) By using the chain rule and the derivative if x>0 (1, -1, if x< O' d (\xl) = dx | d find - (lx3…

A: Just break the mod function in two parts and easily calculate the derivative.

Q: Find the derivative of the function. sin(x) y = /cos(x) In(8 + 3v) dv y'(x) = 0

A: Here we use Leibniz integral rule and define the formula:

Q: Find the derivative. 8 + e y = In 8 + et 8 - ex.

A: Given,          y=ln8+ex8-ex

Q: (a) Find the first derivative of the function y= la) with respect to x by using the chain rule.

A:

Q: Determine the n-th derivative fn) of the function f: (0, +0) → R, f(x) = x ' exp ( 1 - ), n EN

A: f(x)=1x,  f:(0, +∞)→R,…

Q: Compute the derivative of x(x + 2) (2x + 1)(3x + 2) y = (a) directly using the Chain and Product…

A:

Q: Using logarithmic differentiation, Show that y (x)= xa2" e®has no stationary points other than x =…

A:

Q: 19. Determine the derivative of f(x) = e* coshx. O a. e2x O b. ex O c. e4x O d. e5x

A:

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

A:

Q: Let f(x, y,z) = z-sin(xy), x = t, y = In t, z = e4. Find derivative of f directly and by using chain…

A:

Q: Find the derivative of the function. y = e* In x

A: Find the derivative use product rule

Q: 24. Find the first order derivative with respect to x of the function u = log(x+ x .

A: The solution for the above question is as shown below.

Q: Find the derivative: 1+e5 f(x) = = COS 1.

A: The given function is f(x)=cos1+e5x1-e8x

Q: The derivative of f(z) = Log(2iz - 4+6i) at z = -3+i is

A:

Q: Use logarithmic differentiation to find the derivative of y with respect to x. y = ( In (3x))n (9×)…

A: