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If 𝑓 of 𝑥 is equal to two to the power of five 𝑥, what is the value of 𝑓 prime of 𝑥?
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Now 𝑓 prime of 𝑥 is simply the derivative of 𝑓 of 𝑥 with respect to 𝑥.
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So what we need to do is differentiate two to the power of five 𝑥 with respect to 𝑥.
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Now the rule which we can use in order to differentiate exponentials is as follows.
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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.
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Now in our case, we in fact have two to the power of five 𝑥.
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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.
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In order to find 𝑓 prime of 𝑥, we therefore only need to evaluate d by d𝑥 of 32 to the power of 𝑥.
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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.
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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.
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And so now we arrive at our solution.
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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.