Question Video: Finding the First Derivative of an Exponential Function with an Integer Base | Nagwa Question Video: Finding the First Derivative of an Exponential Function with an Integer Base | Nagwa

# Question Video: Finding the First Derivative of an Exponential Function with an Integer Base Mathematics • Third Year of Secondary School

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If π¦ = β3 Γ 2^π₯, determine ππ¦/ππ₯.

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### Video Transcript

If π¦ equals negative three times two to the π₯, determine ππ¦ by ππ₯.

We want to find ππ¦ by ππ₯. 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₯. 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₯.

How do we differentiate the exponential function with base two, two to the π₯? 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₯. Whenever we differentiate an expression, where the variable weβre differentiating with respect to β in our case π₯ β appears in an exponent, then this is a fact we need to use. This means taking any exponential term that we want to differentiate and rewriting it so its base is π.

How do we do this with two to the π₯? Well, we can rewrite two as π to the natural logarithm of two. 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₯. Weβve made the base π. Now, how does that help? Well, we can apply the chain rule.

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§. Now applying the chain rule with π equal to π to the π§, we get negative three times π by ππ§ of π to the π§ times ππ§ by ππ₯.

Letβs clear some space. Here, Iβve just copied the last line of working. What is π by ππ§ of π to the π§? Well, just like π by ππ₯ of π to the π₯ is π to the π₯ or π by ππ of π to the π is π to the π, π by π π§ of π to the π§ is π to the π§. And how about ππ§ by ππ₯? Well, π§ is the natural logarithm of two times π₯. So ππ§ by ππ₯ is just the natural logarithm of two.

Are we done? Well, not quite, we have ππ¦ by ππ₯ in terms of π§ and would really rather prefer it to be in terms of π₯. We can substitute the natural logarithm of two times π₯ for π§. And now, we have our answer written in terms of π₯. But we can do even better.

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₯. Now, we can do the reverse β rewriting π to the natural logarithm of two times π₯ as two to the π₯. And so our final answer is negative three times two to the π₯ times natural logarithm of two.

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. This will probably involve using the natural logarithm function and some laws of exponents. Having written all the exponentials with a base of π, you can then differentiate using the chain rule.

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