Verify that f has an inverse function. Then use the function f and the given real number a to find (f^-1)'(a).  f(x) = cos(4x),     0<x<pi/4,  a = 1   I'm totally lost...

Q: Find the Derivative f(x) = cosh(In(4x))

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Q: Consider the function f(x) whose second derivative is f''(x) = 8x + 6 sin(x). If f(0) = 2 and f'(0)…

A: Given  f''(x) = 8x+6 sin(x),  f(0)=2,  f'(0)=4 We have to find the value of f(3).

Q: Consider the function f(x) whose second derivative is f”(x) = 10x + 8sin(x) If f(0) = 2 and f’(0) =…

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Q: How do I verify d/dx(cos^-1(x))=-1/(sqrt(1-x^2)) using implicite differentiation?

A: Verify the result by implicit differentiation as follows.

Q: Use the General Power Rule, Exponential Rule, or the Chain Rule to compute the derivative. y = (2…

A: Given function is   Differentiating both sides with respect to x, we get

Q: Find the derivative of 2 cos x 1. f(x) x+1

A: As we know that: The Power rule: ddxxn=nxn-1 The Quotient rule: fg'=f'·g-g'·fg2 Given: fx=2cosxx+1

Q: Use logarithmic differentiation to find the derivative of the function. y = (cos(4x))* y'(x) =…

A: In this question we have to find differentiation by using logarithm.

Q: Find the derivative f'(x): f(x)=x4-x2+sin-1x/3 **4-x2 is under a radical sign!

A: Consider the given function f (x) taking (4-x^2) under the radical sign. Differentiate f (x) w.r.t x…

Q: Consider the function f(x) whose second derivative is f"(x) = 3x + 8 sin(x). If f(0) = 2 and f'(0)…

A: Consider the second derivative function, f''x=3x+8sinx Given, f0=2  and f'0=2 We need to find f(x).…

Q: The derivative of In (cos x) is

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Q: What is the first derivative of g(x) = In 4x²+4x+8 ? x 3 tan x

A: we use Logarithmic properties

Q: Find the second derivative of the function 2 f(x) = sin (3x). Show your work.

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Q: Consider the function f(x) whose second derivative is ƒ''(x) = 9x + 6 sin(x). If ƒ(0) = 4 and f'(0)…

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Q: Consider the function f(x) whose second derivative is f''(x) = 2x + 9 sin(x). If f(0) = 4 and f'(0)…

A:

Q: If f(x)=4x(sinx+cosx), f′(x). f′(x) =   Evaluate the derivative at x=5. Enter an approximation,…

A: Given,         fx=4xsinx+cosx

Q: Consider the function f(x) whose second derivative is f''(x) = 4x + 6 sin(x). If f(0) = 3 and f'(0)…

A:

Q: 2sin x -1 2. y = cosx+ 2

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: Calculate the derivative of: cos-(8x) sin-(8x) f(x) f'(x) = 8|

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Q: compute the derivative.  f (x) = (cos2 x)cos x

A: To compute the derivative.  f (x) = (cos2 x)cos x

Q: Consider the function f(x) whose second derivative is f′′(x) = 18x + 3 cos x. If f(0) = 4 and f′(0)…

A:

Q: Find the second derivative of f (x) = 2 sin (3x). Enclose arguments of functions in parentheses. For…

A: Differentiate two times

Q: Consider the function f(x) whose second derivative is f''(x)=9x+9sin(x). If f(0)=2 and f'(0)=3, what…

A:

Q: Consider the function f(x) whose second derivative is f''(x) = 7x + 4 sin(x). If f(0) = 2 and f'(0)…

A:

Q: (1+x²-1) y = tan²¹ / √₁+x²

A:

Q: 3. Find the derivative of f (x) tan (5x + 7), by first principle of differentiation

A: Given function is By first principle of differentiation, we have

Q: Find the 73rd derivative of the function f(x) = 2 sin(x). f (73) (x)

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Q: Solve for the derivative of the following trigonometric functions. 1. f(x) = e3x 2. f(x) = x³ ex 3.…

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Q: Consider the function f(x) whose second derivative is f"(x) = 9x + 4 sin(x). If f(0) = 4 and f'(0) =…

A:

Q: Consider the function f (x) whose second derivative is f"(x) = 10x + 8 sin(x). If f (0) = 2 and…

A: Given function is

Q: Find az when yz+x In y = 2.

A: Use implicit differentiation

Q: Consider the function f(x) whose second derivative is f''(x) = 4x + 3 sin(x). If f(0) = 4 and f'(0)…

A: see the details solution in below

Q: Consider the function f(x) whose second derivative is f"(x) = 9x + 6 sin(x). If f(0) = 2 and f'(0) =…

A:

Q: Find the derivative of f(x) = 9 sinh-' (x²) – 9 In ( Vx4 + 1). %3D

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Q: (A) Find the first derivative of the following function: 1+ cos2x f(x) = %3D 1- cos2x |

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Q: Compute the derivative of each function. f(x) = [sin(4² + x)] [cos(x* + 4)] %3D

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Q: Find derivative of the f(x) = sin-1*- 1 x+1 function

A: To find the derivative of the function

Q: Consider the function f(x) whose second derivative is f''(x) = 5x + 7 sin(x). If f(0) = 2 and f'(0)…

A: Given :f''(x)=5x+7sin(x)

Q: Differentiate the function. f(x) = sin(x) In(3x) sin (x) f '(x) + cos (x) In (3x)

A:

Q: Consider the function f(x) whose second derivative is f''(x) = 7x + 3 sin(x). If f(0) = 3 and f'(0)…

A:

Q: Consider the function f(x) whose second derivative is f''(x) = 9x + 8 sin(x). If f(0) = 2 and f'(0)…

A:

Q: 3. Find if y³ + 2xy - x² = cos x using implicit differentiation.

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Q: Consider the function f(x) whose second derivative is f''(x) = 10x + 8 sin(x). If f(0) = 3 and f'(0)…

A: Given, f''(x)=10x+8sinx(x) After integrating the above equation,…

Q: Consider the function f(x) whose second derivative is f''(x) = 7x + 8 sin(x). If f(0) = 3 and f'(0)…

A: Recall: ∫xn dx=xn+1n+1+c ∫sinaxdx=-cos(ax)a+c ∫cosax dx=sin(ax)a+c ; where 'c' is an integration…

Q: Consider the function f(x) whose second derivative is f''(x) = 3x + 7 sin(x). If f(0) = 2 and f'(0)…

A:

Q: Consider the function f(x) whose second derivative is f" (x) = 9x + 3 sin(x). If f(0) = 3 and f' (0)…

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Q: Find the derivative. Simplify completely. 1. f(x) = 4x³ cos x 2. g(x): Sx-4 In x 3. у sin x 4. f(x)…

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Q: f(x) = In(sin² x)

A: Given,           fx=lnsin2x

Q: Find the function f(x) whose derivatives is sin x and whose graph passes through the point (0,2).…

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Q: Show two differènt ways to find the derivative of the function f (x) = In( and verify that the…

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Q: Consider the function f(x) whose second derivative is f''(x) = 7x + 7 sin(x). If f(0) = 2 and f'(0)…

A: We need to find f(5).

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.