# Video: Evaluating the Derivative of a Power Function

If π(π₯) = 3π₯^(2/3), what is the value of πβ²(27)?

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### Video Transcript

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

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

Applying this formula to π of π₯, we obtain that the derivative of π 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 equals one over π₯ to the power of π for all constants π.

Substituting π₯ equals 27 into this, we obtain that π prime of 27 equals two over 27 to the power of one-third. Now, recall that π to the power of one over π is just another way of writing the πth root of π. 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, π prime of 27 equals two over three, which is our final answer.