Question Video: Finding the Domain, the Derivative, and the Domain of the Derivative of a Radical Function Mathematics • Higher Education

Consider the function 𝑓(π‘₯) = βˆ›π‘₯. a) What is the domain of 𝑓? b) Find an expression for the derivative of our function 𝑓. c) What is the domain of the derivative 𝑓′?

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Video Transcript

Consider the function 𝑓 of π‘₯ is equal to the cube root of π‘₯. Part a), what is the domain of 𝑓?

For part a), we can immediately recall that the cube root of any real number is well-defined on the real numbers. If instead we had a square root, we know that this statement would not be true, since the square root of negative numbers are not defined in the real numbers. As it stands, however, we can answer part a) in a very straightforward manner by saying that the domain of 𝑓 is ℝ, the real numbers.

Moving on to part b), finding the derivative of 𝑓. One of the tools that we have at our disposal is the power rule. This rule tells us that for some function 𝑓 of π‘₯, which takes the form π‘₯ to the power of 𝑛, the derivative of our function would be 𝑛 times π‘₯ to the power of 𝑛 minus one. To apply this to our question, it’s helpful to express the cube root of π‘₯ as π‘₯ to the power of one over three. We can then apply the power rule, multiplying our π‘₯ by one over three, which is our power and subtracting one from the power which gives us negative two over three. An equivalent way to express this would be one over three times the cube root of π‘₯ squared. We’ve now successfully applied the power rule. Snd we’ve sold part b) finding an expression for the derivative of our function 𝑓.

Finally, for part c), finding the domain of this derivative, we used the expression that we just found. For this part of the question, we must consider all points for which 𝑓 dash of π‘₯ is undefined. Since we have 𝑓 dash of π‘₯ in the form of a quotient, we can say that it’ll be undefined when the denominator of this quotient is equal to zero. We, therefore, need to find a value for π‘₯, for which three times the cube root of π‘₯ squared is equal to zero. And the only value which satisfies this is when π‘₯ is equal to zero. Now, since π‘₯ equals zero is the only point for which 𝑓 dash of π‘₯ is undefined over the real numbers, we can say the following. The domain of the derivative of our function 𝑓 dash is the real numbers ℝ minus the set which contains zero.

We have now solved all three parts of our question.

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