Question Video: Finding the First Derivative of a Function Involving Expanding a Square Bracket and Using the Power Rule Mathematics • Higher Education

Find the first derivative of the function 𝑦 = 9𝑥 + 5𝑥(4𝑥² + (5/𝑥))².

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Video Transcript

Find the first derivative of the function 𝑦 equals nine 𝑥 plus five 𝑥 multiplied by four 𝑥 squared plus five over 𝑥 all squared.

The first step with this question is to write it out with both the parentheses in there. That’s cause we have four 𝑥 squared plus five over 𝑥 all squared. So, I’ve written it out so that we have nine 𝑥 plus five 𝑥 multiplied by four 𝑥 squared plus five over 𝑥 multiplied by four 𝑥 squared plus five over 𝑥.

And now what I’m gonna do is I’m gonna rewrite five over 𝑥 in exponent form. And I’m gonna do that using the rule that 𝑥 to the power of negative one is equal to one over 𝑥. So, now in each of the parentheses, we have five 𝑥 to the power of negative one as the last term. So, now what I’m gonna do is expand the parentheses. And to expand the parentheses, I’m gonna use another exponent rule. And that is that if we have 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏, we add the exponents. So, it’s 𝑥 to the power of 𝑎 plus 𝑏.

So therefore, the first term’s gonna be 16𝑥 to the power of four. And that’s cause four multiplied by four is 16. And then 𝑥 squared multiplied by 𝑥 squared gives us 𝑥 to the power of four, as two add two is equal to four. And then we’re gonna get add 20𝑥. That’s cause four 𝑥 squared multiplied by five 𝑥 to the power of negative one. Four multiplied by five is 20.

And then, we got 𝑥 squared multiplied by 𝑥 to the power of negative one. So, two add negative one is just one. So, we just have 𝑥. And then, we have plus another 20𝑥 cause, again, we have positive five 𝑥 to the power negative one multiplied by four 𝑥 squared. And then, finally, we add 25𝑥 to the power of negative two. And that’s cause we have five 𝑥 to the power of negative one multiplied by five 𝑥 to the power of negative one.

So, now the next step is to collect the like terms. We’ve got two 𝑥 terms here, so we can collect them. So, when we’ve done that, we got nine 𝑥 plus five 𝑥 multiplied by 16𝑥 to the power of four plus 40𝑥. And that’s cause we have 20𝑥 plus 20𝑥 plus 25𝑥 to the power of negative two. So, now what we need to do is expand the parentheses by multiplying five 𝑥 by each of the terms inside.

So, the first term is gonna be 80𝑥 to the power of five. So, we’ve got nine 𝑥 plus 80𝑥 to the power of five. And that’s cause we have five 𝑥 multiplied by 16𝑥 to the power of four. Then, plus 200𝑥 squared. And then, finally, plus 125𝑥 to the power of negative one. And that’s because if we have five 𝑥 multiplied by 25𝑥 to the power of negative two. But if we’ve got 𝑥 on its own, that’s the same as 𝑥 to the power of one. So, we’ve got one add negative two, which gives us the power of negative one, or the exponent of negative one.

So, now we rewrite our function in descending exponents of 𝑥. And we’ve got it in a form now that’s easy to differentiate. So, in the first derivative, the first term’s gonna be 400𝑥 to the power of four. And just to remind us how we differentiate, we multiply the exponent by the coefficient, so five by 80, which gives us our 400. And then, we reduce the exponent by one. So, we get five minus one, which gives us four. So, we got 400𝑥 to the power of four.

Then, plus 400𝑥 then plus nine cause if we differentiate nine 𝑥, we just get nine. And then, finally, minus a 125𝑥 to the power of negative two. And it’s minus this because we’re gonna multiply by the exponent, which is a negative one. So, if we multiply 125 by negative one, we get negative 125.

So therefore, we can say that the first derivative of the function 𝑦 equals nine 𝑥 plus five 𝑥 multiplied by four 𝑥 squared plus five over 𝑥 all squared is going to be 400𝑥 to the power of four plus 400𝑥 minus a 125𝑥 to the power of negative two plus nine.

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