Video: APCALC02AB-P1A-Q05-940125304371

If 𝑓(𝑥) = 2^(5𝑥), what is the value of 𝑓′(𝑥)?

01:13

Video Transcript

If 𝑓 of 𝑥 is equal to two to the power of five 𝑥, what is the value of 𝑓 prime of 𝑥?

Now 𝑓 prime of 𝑥 is simply the derivative of 𝑓 of 𝑥 with respect to 𝑥. So what we need to do is differentiate two to the power of five 𝑥 with respect to 𝑥. Now the rule which we can use in order to differentiate exponentials is as follows. We say that, for some constant 𝑎, the differential of 𝑎 to the power of 𝑥 with respect to 𝑥 is equal to 𝑎 to the power of 𝑥 multiplied by the natural logarithm of 𝑎.

Now in our case, we in fact have two to the power of five 𝑥. However, using power rules, we can rewrite this as two to the power of five to the power of 𝑥, which is equal to 32 to the power of 𝑥. In order to find 𝑓 prime of 𝑥, we therefore only need to evaluate d by d𝑥 of 32 to the power of 𝑥.

Applying our rule for differentiating exponentials, where 𝑎 is equal to 32, we can say that this is equal to 32 to the power of 𝑥 multiplied by the natural logarithm of 32. And now we note that we can rewrite 32 to the power of 𝑥 in the slightly simpler form as it was given in the question. And so now we arrive at our solution. And that is that 𝑓 prime of 𝑥 is equal to two to the power of five 𝑥 multiplied by the natural logarithm of 32.

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