Video Transcript
Find the third derivative of the function 𝑦 equals negative 11𝑥 plus 14 over 𝑥.
To find the third derivative of a function, we need to differentiate its equation three times. Before we do that, we may find it helpful to express the second term in our function slightly differently. Using laws of exponents, we can rewrite 14 over 𝑥 as 14𝑥 to the power of negative one. Our equation for 𝑦 then becomes 𝑦 equals negative 11𝑥 plus 14𝑥 to the power of negative one. And we can differentiate each of these terms using the power rule of differentiation. This tells us that for the general power term, 𝑎𝑥 to the 𝑛th power for real values of 𝑎 and 𝑛, its derivative with respect to 𝑥 is equal to 𝑎𝑛𝑥 to the 𝑛 minus oneth power. We multiply by the original exponent 𝑛 and then reduce the exponent by one.
To find the first derivative of our function 𝑦 then, we differentiate once. And you may find it helpful to think of negative 11𝑥 as negative 11𝑥 to the power of one. Applying the power rule to the first term then, we have negative 11 multiplied by one multiplied by 𝑥 to the power of zero. And then, applying the power rule to the second term, we have 14 multiplied by negative one multiplied by 𝑥 to the power of negative two. Now, recall that 𝑥 to the power of zero is simply one. So, the first term is negative 11 multiplied by one multiplied by one, which is just negative 11. Our first derivative therefore simplifies to d𝑦 by d𝑥 equals negative 11 minus 14𝑥 to the power of negative two.
To find the second derivative, we differentiate again. And you may find it helpful if you think of that first term still as negative 11𝑥 to the power of zero. Applying the power rule of differentiation then, we have negative 11 multiplied by zero multiplied by 𝑥 to the power of negative one minus 14 multiplied by negative two multiplied by 𝑥 to the power of negative three. But of course, multiplying by zero just give zero. So, the entire first term is equal to zero. And we see again that the derivative of a constant is just zero. Our second derivative therefore simplifies to 28𝑥 to the power of negative three.
To find the third derivative, we differentiate again, giving 28 multiplied by negative three multiplied by 𝑥 to the power of negative four. This gives negative 84𝑥 to the power of negative four. And finally, we can use our laws of exponents to rewrite this as negative 84 over 𝑥 to the fourth power. So, by differentiating three times and each time applying the power rule of differentiation, we’ve found that the third derivative of our function 𝑦 is negative 84 over 𝑥 to the fourth power.
