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If π¦ equals negative three times two to the π₯, determine ππ¦ by ππ₯.
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We want to find ππ¦ by ππ₯.
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And as π¦ equals negative three times two to the π₯, that means differentiating negative three times two to the π₯ with respect to π₯.
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As the derivative of a number times a function is that number times the derivative of the function, all we have to do now is differentiate two to the π₯ with respect to π₯.
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How do we differentiate the exponential function with base two, two to the π₯?
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Hopefully, we know about the number π, whose special property is that the derivative of π to the π₯ with respect to π₯ is π to the π₯.
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Whenever we differentiate an expression, where the variable weβre differentiating with respect to β in our case π₯ β appears in an exponent, then this is a fact we need to use.
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This means taking any exponential term that we want to differentiate and rewriting it so its base is π.
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How do we do this with two to the π₯?
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Well, we can rewrite two as π to the natural logarithm of two.
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And then using one of our exponent laws, we get that two to the π₯ is π to the natural logarithm of two times π₯.
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Weβve made the base π.
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Now, how does that help?
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Well, we can apply the chain rule.
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If we let π§ equal the natural logarithm of two times π₯, then we need to find negative three times the derivative with respect to π₯ of π to the π§.
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Now applying the chain rule with π equal to π to the π§, we get negative three times π by ππ§ of π to the π§ times ππ§ by ππ₯.
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Letβs clear some space.
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Here, Iβve just copied the last line of working.
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What is π by ππ§ of π to the π§?
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Well, just like π by ππ₯ of π to the π₯ is π to the π₯ or π by ππ of π to the π is π to the π, π by π π§ of π to the π§ is π to the π§.
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And how about ππ§ by ππ₯?
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Well, π§ is the natural logarithm of two times π₯.
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So ππ§ by ππ₯ is just the natural logarithm of two.
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Are we done?
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Well, not quite, we have ππ¦ by ππ₯ in terms of π§ and would really rather prefer it to be in terms of π₯.
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We can substitute the natural logarithm of two times π₯ for π§.
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And now, we have our answer written in terms of π₯.
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But we can do even better.
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Earlier, we showed that we could rewrite two to the π₯ with a base of π as π to the natural logarithm of two times π₯.
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Now, we can do the reverse β rewriting π to the natural logarithm of two times π₯ as two to the π₯.
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And so our final answer is negative three times two to the π₯ times natural logarithm of two.
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In general, if you want to differentiate an expression involving two to the π₯ or three to the π₯ or even as you might see later π₯ to the π₯, you should first write that exponential with a base of π.
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This will probably involve using the natural logarithm function and some laws of exponents.
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Having written all the exponentials with a base of π, you can then differentiate using the chain rule.