Video Transcript
Evaluate the derivative with respect to 𝑥 of root three 𝑥 to the seventh power plus 𝑥 to the ninth power of nine plus six 𝜋.
We see that the function we’ve been asked to differentiate is a polynomial function of 𝑥. There are two key results that we need to recall. Firstly, if we’re differentiating the sum of functions or the sum of terms as we have here, then the derivative with respect to 𝑥 of 𝑓 of 𝑥 plus 𝑔 of 𝑥 is equal to the derivative with respect to 𝑥 of 𝑓 of 𝑥 plus the derivative with respect to 𝑥 of 𝑔 of 𝑥. Essentially, we can differentiate each term separately and add the derivatives together.
Secondly, we recall the power rule for differentiation. For any real number 𝑛, the derivative with respect to 𝑥 of 𝑥 to the 𝑛th power is equal to 𝑛 multiplied by 𝑥 to the power of 𝑛 minus one. We multiply by the original power and then decrease the power by one. A special case of this is that the derivative with respect to 𝑥 of some constant 𝑐 is just zero, which we can understand if we think of 𝑐 as 𝑐 multiplied by 𝑥 to the power of zero. When we differentiate, we’ll be multiplying by that power zero and therefore our answer will be zero.
Finally, we also recall that if we’re finding the derivative with respect to 𝑥 of some constant 𝑎 multiplied by 𝑓 of 𝑥, then this is equal to 𝑎 multiplied by the derivative with respect to 𝑥 of 𝑓 of 𝑥. We can differentiate our function and then multiply by the constant 𝑎. Let’s use all of these rules, then, to evaluate the derivative we’ve been asked to find.
The first rule tells us that we can just differentiate each term separately. So let’s begin with root three 𝑥 to the seventh power. By the second rule, differentiating 𝑥 to the seventh power will give seven 𝑥 to the sixth power. And by our final rule, we can then just multiply it by that constant of root three. Then we differentiate the second term. The derivative of 𝑥 to the ninth power by our second rule is nine 𝑥 to the eighth power. And by the fourth rule, we can multiply by that factor of one ninth. Finally, we differentiate six 𝜋 which recall is just a constant. And therefore, its derivative with respect to 𝑥 will just be zero.
We can cancel a factor of nine in that second term to leave us just with 𝑥 to the eighth power. And the first term simplifies to seven root three 𝑥 to the sixth power. So by recalling these key rules of differentiation, most importantly the power rule. We’ve found that the derivative with respect to 𝑥 of this polynomial function is equal to seven root three 𝑥 to the sixth power plus 𝑥 to the eighth power.
