Video: Differentiating a Combination of Exponential and Rational Functions Using the Chain and the Quotient Rules

Find the derivative of the function 𝑓(𝑧) = βˆ’3𝑒^(4𝑧/(4𝑧 + 1)).

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Video Transcript

Find the derivative of the function 𝑓 of 𝑧 equals negative three 𝑒 to the four 𝑧 over four 𝑧 plus one.

Here we have a function of a function or a composite function. This tells us we’re going to need to apply the chain rule. This says that the derivative of 𝑓 of 𝑔 of π‘₯ is equal to the derivative of 𝑓 of 𝑔 of π‘₯ multiplied by the derivative of 𝑔 of π‘₯. Alternatively, we can say that if 𝑦 is equal to this composite function, 𝑓 of 𝑔 of π‘₯, then if we let 𝑒 be equal to 𝑔 of π‘₯, then 𝑦 is equal to 𝑓 of 𝑒. And this means we can say that the derivative of 𝑦 with respect to π‘₯ is equal to the derivative of 𝑦 with respect to 𝑒 multiplied by the derivative of 𝑒 with respect to π‘₯.

In this example, we say that 𝑦 is equal to negative three times 𝑒 to the power of 𝑒, where 𝑒 is equal to four 𝑧 over four 𝑧 plus one. And since we’re going to be differentiating with respect to 𝑧, we alter the formula slightly. And we say that d𝑦 by d𝑧 is equal to d𝑦 by d𝑒 times d𝑒 by d𝑧. So we’re going to need to work out d𝑦 by d𝑒 and d𝑒 by d𝑧. d𝑦 by d𝑒 is a fairly easy one to differentiate. We know that the derivative of 𝑒 to the power of 𝑒 is 𝑒 to the power of 𝑒. So the derivative of negative three 𝑒 to the power of 𝑒 is negative three 𝑒 to the power of 𝑒. And then, we can replace 𝑒 with four 𝑧 over four 𝑧 plus one to get negative three 𝑒 to the four 𝑧 over four 𝑧 plus one. But what about the derivative of four 𝑧 over four 𝑧 plus one?

Well, here, we need to use the quotient rule. This says that the derivative of 𝑓 of π‘₯ over 𝑔 of π‘₯ is equal to the function 𝑔 of π‘₯ times the derivative of 𝑓 of π‘₯ minus the function 𝑓 of π‘₯ times the derivative of 𝑔 of π‘₯. And that’s all over 𝑔 of π‘₯ squared. We change 𝑓 of π‘₯ to 𝑓 of 𝑧 and 𝑔 of π‘₯ to 𝑔 of 𝑧. Then, the derivative of the numerator of our fraction is four. And the derivative of the denominator of our fraction is also four. So the equivalent to 𝑔 of 𝑧 times the derivative of 𝑓 of 𝑧 is four 𝑧 plus one times four. And the equivalent to 𝑓 of 𝑧 times the derivative of 𝑔 of 𝑧 is four 𝑧 times four. And that’s all over the denominator squared. That’s four 𝑧 plus one squared. Now, distributing the parentheses and we simply end up with four on the numerator of this fraction. So d𝑒 by d𝑧 is equal to four over four 𝑧 plus one squared.

Let’s now substitute everything we have into the formula for the chain rule. d𝑦 by d𝑒 times d𝑒 by d𝑧 is negative three 𝑒 to the four 𝑧 over four 𝑧 plus one times four over four 𝑧 plus one squared. And if we simplify, we see that the derivative of our function is negative 12𝑒 to the four 𝑧 over four 𝑧 plus one all over four 𝑧 plus one squared.

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