The method of u-substitution is the integral analog of the chain rule of derivatives and the method of the integration of parts is the integral analog of the product rule for derivatives. How do you find the relationship between the derivative and their ananlogous integral rules?

Q: give a problem and solution for differentiation rule of exponential and logarithmic functions in…

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

Q: Solve for the derivative of x² – y² = 9 using implicit differentiation %3D

A: Given :                x2 - y2 = 9To find :                 y'

Q: Find the derivative: y = ln² (3x + 1) O y' = O O O y' = y' = y = 2 ln(3x + 1) (3x + 1) 6 (3x + 1) 3…

A: Consider the function y=ln2(3x+1). The objective is to find the derivative of the function.

Q: Find the derivative of the given function. Infsech(x) + tanh(x)) 1) sechx+tanhr sinhx+1 sechx-tanhx…

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Q: Find the derivatives of the functions in Exercis y = xe-x + ex3

A: y=xe-x+ex3

Q: Find the first derivative: y= ln v5x3 –- 2 5, O y = 222 %3D 523-2 O y' = 322 5z3-2 O y' = 622 5z3-2…

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Q: Find the derivative of the function f(x) = 2sin x %3D

A: Given       f(x)=2sin x

Q: HiHtar is THe secoyd derivative OF Y

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Q: 2. If the derivative of an accurately determined function is replaced with a finite difference, how…

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

Q: Determine the second derivative of f (x)= x 'e* at x = -2 with a step-size of h=0.50 using Central…

A: We have to find the second derivative of the following function at x=-2 with step size h=0.5 using…

Q: Explain the derivatives and integration.

A: Solution: Derivatives: Definition of derivatives is , the rate of change of a quantity with respect…

Q: ) y = In³ )) (tanh²(scc '(x²)

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Q: Find the points on the graph of f where the tangent line is horizontal. tangent line is 3x^2-16…

A: The tangent line is horizontal where y'=0

Q: Solve for the derivative of the Hyperbolic Differentiation. 1. y = ex cos h 2x 2. y = ln sin h x…

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Q: 2. Determine the second derivative of f(x)= x'e* at x = -2 with a step-size of h=0.50 using Central…

A: Given function is f(x) = x2e2x and h = 0.50

Q: By using the second derivative test for functions of two variables, find three positive numbers…

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Q: Find the first derivative of f(x) = e3r

A: To find the first derivative of fx=x3e3x. Solution: Quotient rule of derivative states that:…

Q: Find the derivative of the function using 3-step rule: v (t) = -.

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Q: What is the first derivative of f(x) = ln 3z²+7z ?

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Q: How can you relate antiderivatives as an opposite of derivatives in your life? explain

A: Solution is given below:

Q: y=C1x + C2e + x2

A: The givne equation is:y=C1x+C2e-x+x2To eliminate the arbitrary constant/s

Q: 8. Find the derivative of y = x arctan 2.x --In(1+4x). -- - 4.

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Q: To investigate these values, we find the second derivative f"(x) = %3D At the positive critical…

A: Given function is : fx=3x-4x3

Q: What is the connection of the concept of limits to slope to derivatives (delta method to simple…

A: The limits of the function at a point are calculated using the formula . The limits are used to…

Q: How would you find the stationary points of partial derivative equations coming from a multivariate…

A: Stationary points:

Q: maximum point of y x – 2

A: Condition for Extreme Value: A continuous function fx, y has extrema at a, b if fx=fy=0 at a, b…

Q: Find  y ″  by implicit differentiation. 4x3 − 5y3 = 8

A: First we find y' by differentiating  both sides of the given equation. Then find y"  by using the…

Q: Shows the derivative function of: TC = Q³ + 2Q²  + FC(Q), where TC = Total Cost and FC = Fixed…

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Q: can explain for me what is Applications of Partial Derivative ?

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Q: Which derivative rule is used to derive the Integration by Parts formula?

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Q: Find the first derivative of y = 1 using the Chain Rule. (1– x4)6

A: These questions are related to differentiation, We will solve them using given information.

Q: Please kindly use chain rule on the below to find the derivative and label all parts.  h(x)= cos…

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Q: Find the derivative of x2cosx+2xsin x O 2xcos x O x² sin x + 2xcos x Ọ xzcosx y=x²sinx

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Q: How can you relate antiderivatives as an opposite of derivatives in your life?

A: Solution:   Antiderivatives are the inverse operations of derivatives or the backward operation…

Q: Find the derivative of xn by first principle or ab-initio method.

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Q: Find the derivative of y given that y= x In x et

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Q: What is the difference of antidifferentiation and anti-derivative?

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Q: Find the critical value of f (x) = vx² + x – x using the second derivative test. %3D

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Q: What is the rule for calculating the derivative of a composite of two differentiable functions? How…

A: Let us we consider a function which is real valued function which is composite of of two functions…

Q: y = Vx* X

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Q: On which open interval is the derivative positive/ negative .

A: Given:

Q: How do you find the partial derivative of a given function using the definition formula?

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Q: Solve for the derivative of the function using chain rule. 14. f(x)= 3lnx 15. f(x)= cos(cosx))

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Q: Explain how to find a first partial derivative of a function oftwo variables.

A: To get the first partial derivative of a function z = f(x, y) with respect to x we have to…

Q: Determine the second derivative of f(x) = x²e²x at x = -2 with a step-size of h=0.50 using Central…

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Q: What is something interesting about integrationa and derivatives? Why is it interesting?

A: Given What is something interesting about integration and derivatives? Why is it interesting?

Q: Find the fourth derivative of the given: (Show complete solution from first to  fourth derivative)…

A: The function is given by y = (2x+1)25

Q: What is the 2nd derivative of y=cosh(x+y)?

A:

Q: 5. Find the 25" derivative of y= In x.

A: We have to find the 25th derivative of y=ln(x)

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: