How is the natural logarithm function defined as an integral? What are its domain, range, and derivative? What arithmetic properties does it have? Comment on its graph.

Q: Evaluate the definite integral

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Q: Find the indefinite integral and check the result by differentiation.

A: Let,f(x) = x5+1               ......(1)And the given integral be y - y = ∫x5+1 dxy =…

Q: The Differentiation to be made must be based on the following topics. - Logarithmic Fuction Give 3…

A: Let us give three examples of logarithmic functions and find their differentiation in the next…

Q: Evaluate the definite integral.

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Q: Find the indefinite integral and check the result by differentiation. 3 3 dx (5 – 9x2

A: Given               ∫x(5-9x2)3 dx

Q: Use logarithmic differentiation to find the derivativeof the function.

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Q: Using Logarithmic differentiation, find F(x) F(x)=x8/ex(x-1)

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Q: What is the derivative of the natural logarithm function ln x? How does the domain of the derivative…

A: Given: fx=ln x To find: (a) The derivative of f(x)  (b) Compare the domain of the derivative with…

Q: Use the same-derivative argument, as was done to prove the Product and Power Rules for logarithms,…

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Q: Find the indefinite integral and check the result by differentiation. (5 – 4x²) dx

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Q: Find the indefinite integral and check the result by differentiation.

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Q: Find the indefinite integral and check the result by differentiation.

A: To find the integral

Q: Use logarithmic differentiation or an alternativemethod to find the derivative of the function  y =…

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Q: 17. x dx

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Q: Find the indefinite integral and check the result by differentiation

A: Integration is summation of discrete data. The integral is calculated for the functions to find…

Q: Compute for the derivative of the function using logarithmic differentiation. r(θ) = (cscθ)ℼ^θ

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Q: Find the indefinite integral and check the result by differentiation.

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Q: Find the indefinite integral and check the result by differentiation.

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Q: differentiate a function by logarithmic differentiation

A: take natural logarithm to both sides

Q: Find the indefinite integral of the following function and check the result by differentiation.

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Q: Differentiate  f(x)  = log 10  ( 2+ sinx )

A: Rewrite the given function as shown below: fx=log102+sinxfx=ln2+sinxln10        since logab=lnblna

Q: Use LOGARITHMIC differentiation to differentiate the function Simplify where necessary

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Q: B. EVALUATE EACH INDEFINITE OR DEFINITE INTEGRAL. DO NOT SIMPLIFY YOUR RESULT. (12-sec"(x) dx 21. (…

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Q: What are the formulas for differentiation and integration? How are they applied?

A: To write: Formulas for differentiation and integration.

Q: Ose logarithmic differentiation of an alterativt methind to findthe derivative of the function?

A: We will use logarithmic differentiation to find the derivative of the given function

Q: tan x dx = In|sec x|+ C

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Q: Find the indefinite integral and check the result by differentiation. Step 1

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Q: Find the indefinite integral and check the result by differentiation.

A: Integration is summation of discrete data. The integral is calculated for the functions to find…

Q: Find the indefinite integral and check the result by differentiation. + sece) de

A: We need to integrate ∫θ4+sec2θdθ

Q: Find the indefinite integral and check the result by differentiation. ∫3√x2 dx

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Q: Find the indefinite integral and check the result by differentiation.

A: ∫7x1-x23dxTake the constant out:  ∫a·fxdx=a·∫fxdx=7·∫x1-x23dxSubs u = 1-x2=7·∫ -12u3duTake the…

Q: Find the indefinite integral and check the result by differentiation                         ∫ 6x2 /…

A: Consider the following integral: ∫6x24x3-93dx Substitute 4x3-9=u→12x2dx=du→6x2dx=12du in the above…

Q: Find the indefinite integral and check the result by differentiation.

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Q: Find the indefinite integral and check the result by differentiation. 10x 5 -

A: To find the integral we use integration by substitution method.

Q: Evaluate the definite integral: x – 1

A: I am going to solve the given problem by using some simple calculus to get the required result of…

Q: Find the indefinite integral and check the result by differentiation. ∫(x + 7) dx

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Q: How is ln x affected if x is multiplied by et?

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Q: Find the indefinite integral and check the result by differentiation.

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Q: Find the indefinite integral and check the results by differentiation 4x² + 3)²dx

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Q: Use Logarithmic differentiation to evaluate f'(x) f(x)=(1+x2)sinx Thanks!

A: To find the derivative of the function  f(x) = (1 + x2)sinx

Q: Find the indefinite integral and check the result by differentiation                     ∫x3(x4 +…

A: ∫x3x4+32 dx Put, x4+3=t⇒4x3+0=dtdx⇒4x3 dx=dt⇒x3 dx=dt4 Therefore, ∫x3x4+32 dx=∫t2 dt4=14∫t2…

Q: Explain briefly how we can compute the derivative of a function y=f(x) using logarithmic…

A: Logarithmic differentiation is an important method of differentiation. To find the derivative of the…

Q: logarithmic

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Q: (1-x²)-1/2 Arcsin x

A: We will integrate the given function using simple Integration method.

Q: Find the indefinite integral and check the result by differentiation.            ∫(9x8 − 2x − 6) dx

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Q: Give one problem and solution for each differentiation rule of exponential functions.

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Q: Apply logarithmic differentiation given the ff: In x

A: y=xln xtaking ln on both sides ln y=ln xln xlny=ln(x)*ln(x)         ln xa=alnxln y=(ln(x))2…

Q: Apply logarithmic differentiation given

A: Given query is to find derivative using logarithm.

Q: Find the indefinite integral and check the result by differentiation.              ∫(13 − x) dx

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Q: Find the indefinite integral and check the result by differentiation                     ∫ x /√(1 −…

A: Given indefinite integral ∫x1-x2dx Substitute the value of 1-x2=t and differentiate it w.r.t "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: