Differentiate the following by using "limit"
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Step-by-step explanation:

we want to differentiate the following by using limit:

derivative definition by limit given by

given that,

f(x)=√x

so,

f(x+∆x)=√(x+∆x)

thus substitute:

multiply both the numerator and denominator by the conjugate of the numerator:

simplify which yields:

simplify square:

collect like terms:

reduce fraction:

get rid of ∆x as we are approaching its to 0

simplify addition:

and we are done!

the solution of the equation is 2.23 ⇒ answer b

step-by-step explanation:

* lets revise the meaning of exponential function

- the form of the exponential function is y = ab^x, where a ≠ 0, b > 0 ,  

  b ≠ 1, and x is any real number

- it has a constant base b

- it has a variable exponent x

- to solve an exponential equation, take the log or ln of both sides,  

  and solve for the variable

* lets solve the problem

∵

- the base is 7 and the exponent is x

- insert ㏑ in both sides

∴

- use the rule

∴ x ㏑(7) = ㏑(77)

- to find x divide both sides by ㏑(7)

∴

* the solution of the equation is 2.23 to the nearest 2 decimal places