Question

find dy for y = 9x^7 - 3√{3x}. dy =

Explanation:

Step1: Differentiate \(9x^7\)

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(y_1=9x^7\), \(y_1^\prime=9\times7x^{7 - 1}=63x^6\)

Step2: Rewrite and differentiate \(-3\sqrt{3x}\)

Rewrite \(-3\sqrt{3x}=-3\sqrt{3}x^{\frac{1}{2}}\). Using the power rule \((x^n)^\prime = nx^{n - 1}\), \(y_2^\prime=-3\sqrt{3}\times\frac{1}{2}x^{\frac{1}{2}-1}=-\frac{3\sqrt{3}}{2}x^{-\frac{1}{2}}=-\frac{3\sqrt{3}}{2\sqrt{x}}\)

Step3: Find \(dy\)

Since \(dy = y^\prime dx=(y_1^\prime + y_2^\prime)dx\), \(dy=(63x^6-\frac{3\sqrt{3}}{2\sqrt{x}})dx\)

Answer:

\(dy=(63x^6-\frac{3\sqrt{3}}{2\sqrt{x}})dx\)