Video: Evaluating the Derivative of a Power Function

If 𝑓(π‘₯) = 3π‘₯^(2/3), what is the value of 𝑓′(27)?

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Video Transcript

If 𝑓 of π‘₯ equals three times π‘₯ to the power of two over three, what is the value of 𝑓 prime evaluated at 27?

In order to evaluate 𝑓 prime of 27, we will first compute the derivative of 𝑓 with respect to π‘₯, 𝑓 prime of π‘₯. We will then substitute π‘₯ equals 27 into it. Notice that 𝑓 of π‘₯ is a function of the form π‘Žπ‘₯ to the power of 𝑏, where π‘Ž equals three and 𝑏 equals two over three. Recall that in order to differentiate such functions with respect to π‘₯, we multiply the coefficient π‘Ž by the exponent 𝑏 and decrease the exponent by one.

Applying this formula to 𝑓 of π‘₯, we obtain that the derivative of 𝑓 with respect to π‘₯ is three times two over three times by π‘₯ to the power of two over three minus one. This simplifies to two times π‘₯ to the power of negative one-third. We can rewrite this as two over π‘₯ to the power of one-third using the fact that π‘₯ to the power of negative π‘Ž equals one over π‘₯ to the power of π‘Ž for all constants π‘Ž.

Substituting π‘₯ equals 27 into this, we obtain that 𝑓 prime of 27 equals two over 27 to the power of one-third. Now, recall that π‘Ž to the power of one over 𝑏 is just another way of writing the 𝑏th root of π‘Ž. And so, 27 to the power of one-third equals the third root of 27. This is equal to three as three times three times three equals 27. So, 𝑓 prime of 27 equals two over three, which is our final answer.

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