Video: Evaluating an Expression Containing Derivatives of a Root and Polynomial Functions Using the Chain Rule

Given that 𝑦 =√(4π‘₯Β² βˆ’ 5) and 𝑧 = 5π‘₯Β² + 9, determine (𝑦(d𝑦/dπ‘₯)) + (d𝑧/dπ‘₯).

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Video Transcript

Given that 𝑦 is equal to the square root of four π‘₯ squared minus five and 𝑧 is equal to five π‘₯ squared plus nine, determine 𝑦 multiplied by the derivative of 𝑦 with respect to π‘₯ plus the derivative of 𝑧 with respect to π‘₯

The question gives us an equation for 𝑦 in terms of π‘₯ and an equation for 𝑧 in terms of π‘₯. And the expression we are asked to evaluate only has derivatives with respect to π‘₯, so we can attempt to do this directly. To help us calculate this, we recall some derivative rules. First, by using the chain rule we have that the derivative of a function 𝑓 raised to the 𝑛th power is equal to 𝑛 multiplied by 𝑓 prime of π‘₯ multiplied by 𝑓 of π‘₯ raised to the power of 𝑛 minus one. And second, we have a particular case of this rule. The derivative of π‘₯ raised to the 𝑛th power is 𝑛 multiplied by π‘₯ to the power of 𝑛 minus one.

So, let’s start by calculating the derivative of 𝑦 with respect to π‘₯. We have that this is equal to the derivative of the square root of four π‘₯ squared minus five with respect to π‘₯. And we can notice that taking a square root is equivalent to raising something to the power of a half. What this means is we’re differentiating the function of the form 𝑓 of π‘₯ raised to the 𝑛th power, where 𝑛 is equal to a half, and 𝑓 of π‘₯ is equal to four π‘₯ squared minus five.

So, to differentiate this, we use our derivative rule. First, we multiply it by 𝑛, which is a half. Next, we multiply this by the derivative of our function 𝑓 of π‘₯, which is four π‘₯ squared minus five. And finally, we multiply this by 𝑓 of π‘₯ raised to the power of 𝑛 minus one. So, we get four π‘₯ squared minus five all raised to the power of a half minus one. We can simplify this expression by first calculating the derivative of four π‘₯ squared minus five with respect to π‘₯, which we can calculate by using our derivative rule to just be eight π‘₯.

And we can also simplify our exponent, since a half minus one is just equal to negative one-half. This gives us a half multiplied by eight π‘₯ multiplied by four π‘₯ squared minus five raised to the power of negative a half. Next, we can simplify a half multiplied by eight π‘₯ to be four π‘₯. And instead of multiplying by four π‘₯ squared minus five raised to the power of negative a half, we can divide by this raised to the power of positive one-half. This gives us that the derivative of 𝑦 with respect to π‘₯ is equal to four π‘₯ divided by four π‘₯ squared minus five raised to the power of a half.

Now, let’s calculate the derivative of 𝑧 with respect to π‘₯. So, that’s the derivative of five π‘₯ squared plus nine with respect to π‘₯. We can calculate this derivative directly using our power rule for differentiation. This gives us that derivative of 𝑧 with respect to π‘₯ is equal to 10π‘₯. We are now ready to evaluate the expression given to us in the question. That is 𝑦 multiplied by the derivative of 𝑦 with respect to π‘₯ plus the derivative of 𝑧 with respect to π‘₯.

We start by substituting in that 𝑦 is equal to the square root of four π‘₯ squared minus five. Next, we substitute in the expression we calculated for the derivative of 𝑦 with respect to π‘₯. So, that’s four π‘₯ divided by four π‘₯ squared minus five raised to the power of a half. And then, we substitute in our expression for d𝑧 by dπ‘₯, which in this case is 10π‘₯. We can then notice that our denominator, four π‘₯ squared minus five all raised to the power of a half, is actually equal to the square root of four π‘₯ squared minus five.

So, we can cancel this shared factor in our numerator and our denominator, giving us four π‘₯ plus 10π‘₯, which we can evaluate to give us 14π‘₯. Therefore, we have shown that if 𝑦 is equal to the square root of four squared minus five, and 𝑧 is equal to five π‘₯ squared plus nine. Then the expression 𝑦 multiplied by the derivative of 𝑦 with respect to π‘₯ plus the derivative of 𝑧 with respect to π‘₯ is actually equal to 14π‘₯.

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