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)