Consider the following function. g(x) = 3x3 - 3x Find the first and second derivatives of the function. g'(x) = 9x- 3 g"(x) = 18x Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) (-00, – 0.6) U (0.6,00) × concave upward concave downward

Q: Find derivatives of the function defined as follows. y = -4e-0.3x

A: Given,  y=−4e−0.3xFormula→ddxaex=aex  where a is constant.

Q: Q-1- Find the derivatives of the functionS 1- Y+In (xy) = |

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Q: 1. Find the derivative of each of the following functions using chain rule: y=3x* (x* -2) a)

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Q: Find the derivative of the function. y = tan an-¹ (√x-4)

A: We can find the derivative using chain rule as below.

Q: 1. Find the second derivative of the function. 8(x) = 3\/x + e7*In(x)

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Q: Find the derivative of the function.

A: We are given with the function g(x) and need to find the derivatives

Q: Find the derivative of the function g(x) = x3/e2x

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Q: Please I want solution of question 7 with step by step. Many Thanks

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Q: Find the 80th derivative of y = cos(2x).

A: The given function is y=cos(2x)

Q: - 3x Find the derivative of y with respect to x if y = e

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Q: Find the derivative of the function. g(x) = e-X + 3e2x - 2e-x + 11 g'(x) =

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Q: Find the derivative of the function. y = In(e7x + 5)

A: Given y=ln(e7x+5)

Q: Solve for the first, second, and third derivatives of y = 1/(4 + x) and then find a formula for its…

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Q: 2) Let y = xe* and find the first derivative y'. Where is y'>0?What does this tell you about the…

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Q: II. Find the third derivative of the following: 2x-7 1. y = 2x+11

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Q: Find the derivatives of the functions y = (3 + cos3 3x)-1/3

A: Given that, y=3+cos33x-13

Q: Find the first derivative of: x = t²e2t

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Q: Find the derivative of the function.

A: Some important derivative formulas: 1. d(lnx)/dx = 1/x 2. d(√x)/dx = 1/(2√x) 3. d[N/D]/dx = [D*(N)'…

Q: 1. Find the second derivative of the following functions with respect to x. y = 3x2 + 5x - 1   2.…

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Q: Find the derivative of the function. y = e-9/x

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Q: Find the derivative of the function.s(x) = e3x − 1/x3 − 1

A: The given function is

Q: Find the derivatives of the function defined as follows. y = -3e2sin x

A: The function: y=-3 e2 sin x

Q: f. y=In(x8+1)

A: We will solve the problem

Q: Find the derivative of the function. y = e In(7x)

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Q: Q1:- Find the first order derivative of each of Inx a) y = e*secx + CsCx

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Q: B) Find derivative of the following function d -6tanx dx

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Q: Find the derivative of the function.

A: First, we expand using properties of log

Q: Find the derivative of the function.

A: To find derivative of the function y=e2xe2x+1. Solution: Quotient rule of differentiation states…

Q: y = In (*+1) °(x+1), (x+2)3

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Q: 1. Find ALL THE DERIVATIVES of the function: f (x) = 2x7 – a5 + 5a³ – 8x + 4 2. Use logarithmic…

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Q: 4. Find the derivative of the functions. (2x2-3) a. y = (x247)5 sin x b. y = 1+cosx

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Q: 11 8 Find the derivative of the function: y = 3(x – 4)' x-4

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Q: 3. Find the second derivative of f (x) = e²/*° and fully simplify your answer.

A: fx=e2x3

Q: 3. Find all possible functions with the given derivative y' =- 2

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Q: 3x-4 1. Y= Arctan 4

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Q: Find the 85th derivative of y = cos(3x).

A: Do the derivative many times and observe the pattern that derivative follows. Generalized it and…

Q: 2x Find the derivative of h(x) 2x + 3

A: Given:  Function  h(x)=-2x2x+32

Q: -6-X 1.) y = cos -6.x3 y =

A: We have to find the derivative of the function: y=cos-1 -6x3 We know the formula of derivatives,…

Q: Find the derivatives of the functions y = (1 + 2x)e-2x

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Q: Find the derivative of the function. y= -3(6x2 +8) -6 dy dx (Type an expression using x as the…

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Q: Find the derivative of the function. f(x)=4tan-1(e9x)

A: Consider the given equation

Q: Find derivatives of the functions defined as follows: y = e-2x

A: The derivative of the function ex is ddxex=ex. Find the derivative of the function y=e-2x using…

Q: Find the first derivative of the following implicit function: X² + 3xy + y² = 4

A: Given : x2+3xy+y2=4 Lets take first derivative  Take derivative with respect to x. apply the power…

Q: f(x) = e²x cos(x)

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Q: Find the first derivative of the following implicit function. x² + 3xy + y² = 4

A: Given : x2+3xy+y2 = 4

Q: at x= 1

A: 9). Given function is fx=2ex-x. To find derivative of the given function at x=1. Solution:…

Q: f(x) = e²x cos(x) e2

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Q: 1. Find the first derivative of the following. а. у%33x y = 3x² + 6 y = 2-1

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Q: Find the derivatives of following functions e2x f(x) 1- ex

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: 7. Find the second derivative of the following function: y = e** +In(7e¯* + 5x³) – 5²× +-

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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.