If 𝑓 of 𝑥 equals three times 𝑥 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 𝑥, 𝑓 prime of 𝑥. We will then substitute 𝑥 equals 27 into it. Notice that 𝑓 of 𝑥 is a function of the form 𝑎𝑥 to the power of 𝑏, where 𝑎 equals three and 𝑏 equals two over three. Recall that in order to differentiate such functions with respect to 𝑥, we multiply the coefficient 𝑎 by the exponent 𝑏 and decrease the exponent by one.
Applying this formula to 𝑓 of 𝑥, we obtain that the derivative of 𝑓 with respect to 𝑥 is three times two over three times by 𝑥 to the power of two over three minus one. This simplifies to two times 𝑥 to the power of negative one-third. We can rewrite this as two over 𝑥 to the power of one-third using the fact that 𝑥 to the power of negative 𝑎 equals one over 𝑥 to the power of 𝑎 for all constants 𝑎.
Substituting 𝑥 equals 27 into this, we obtain that 𝑓 prime of 27 equals two over 27 to the power of one-third. Now, recall that 𝑎 to the power of one over 𝑏 is just another way of writing the 𝑏th root of 𝑎. 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.