(a) Given the function f(x) = aª for some real constant a. %3D i) Use the properties of natural logarithm and exponential functions to find f'(x) = a" M(a) and write M(a) as a natural logarithm. ii) Use the definition of derivative to show that f'(x) = a"L(a) and write an expression for L(a) as a limit. iii) Prove that M(a) is the limit of L(a).

Q: 3. Consider the function +R where f(r)= log,(2-52). (a) Find the rule for f-. (b) State the range of…

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Q: Use logarithmic differentiation to expand and differentiate y =(x+4)8 / sqrt x4+9

A: we have given y=x+48x4+9now apply log both sides ,we getlog y=logx+48x4+9=8log(x+4)-12log(x4+9)now…

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Q: f (x) = sec(3x – 1) * log3(2 +x) -

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A: Given:

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Q: How to use a graph of a derivative to determine the parent function.

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A: Follow step 2 and 3 .

Q: Find the derivative using logarithmic.   sq root of ((x2-1)/(x2+1))          for x>1

A: Here given for x>1

Q: Use the following graph of the function f(x) to draw f'(x) and f''(x).

A: Given: Graph of the function f(x) is shown as:

Q: Determine the derivative: y = log (3x2) (01)प य log (3) ** In(10) 6 log (Sx) r In(10) Oy= 18x In(10)…

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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: