find dy for y = 7x^8+5\\sqrt{5x}.\ndy = □

Explanation:

Step1: Rewrite the square - root term

Rewrite $y = 7x^{8}+5\sqrt{5x}$ as $y = 7x^{8}+5\sqrt{5}\cdot x^{\frac{1}{2}}$.

Step2: Apply the power rule for differentiation

The power rule states that if $y = ax^{n}$, then $y^\prime=\frac{dy}{dx}=nax^{n - 1}$. For the first term $y_1 = 7x^{8}$, $\frac{dy_1}{dx}=8\times7x^{8 - 1}=56x^{7}$. For the second term $y_2 = 5\sqrt{5}x^{\frac{1}{2}}$, $\frac{dy_2}{dx}=\frac{1}{2}\times5\sqrt{5}x^{\frac{1}{2}-1}=\frac{5\sqrt{5}}{2}x^{-\frac{1}{2}}$.

Step3: Find $dy$

Since $dy=\frac{dy}{dx}dx$, we have $dy=(56x^{7}+\frac{5\sqrt{5}}{2\sqrt{x}})dx$.

Answer:

$(56x^{7}+\frac{5\sqrt{5}}{2\sqrt{x}})dx$