# Question Video: Differentiating Polynomials Mathematics • Higher Education

Evaluate d/dπ₯ (β(3)π₯β· + (π₯βΉ/9) + 6π).

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### Video Transcript

Evaluate the derivative with respect to π₯ of root three π₯ to the seventh power plus π₯ to the ninth power of nine plus six π.

We see that the function weβve been asked to differentiate is a polynomial function of π₯. There are two key results that we need to recall. Firstly, if weβre differentiating the sum of functions or the sum of terms as we have here, then the derivative with respect to π₯ of π of π₯ plus π of π₯ is equal to the derivative with respect to π₯ of π of π₯ plus the derivative with respect to π₯ of π of π₯. Essentially, we can differentiate each term separately and add the derivatives together.

Secondly, we recall the power rule for differentiation. For any real number π, the derivative with respect to π₯ of π₯ to the πth power is equal to π multiplied by π₯ to the power of π 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 π₯ of some constant π is just zero, which we can understand if we think of π as π multiplied by π₯ to the power of zero. When we differentiate, weβll be multiplying by that power zero and therefore our answer will be zero.

Finally, we also recall that if weβre finding the derivative with respect to π₯ of some constant π multiplied by π of π₯, then this is equal to π multiplied by the derivative with respect to π₯ of π of π₯. We can differentiate our function and then multiply by the constant π. Letβs use all of these rules, then, to evaluate the derivative weβve been asked to find.

The first rule tells us that we can just differentiate each term separately. So letβs begin with root three π₯ to the seventh power. By the second rule, differentiating π₯ to the seventh power will give seven π₯ 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 π₯ to the ninth power by our second rule is nine π₯ 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 π₯ will just be zero.

We can cancel a factor of nine in that second term to leave us just with π₯ to the eighth power. And the first term simplifies to seven root three π₯ to the sixth power. So by recalling these key rules of differentiation, most importantly the power rule. Weβve found that the derivative with respect to π₯ of this polynomial function is equal to seven root three π₯ to the sixth power plus π₯ to the eighth power.