find dy for y = 5x^4 - 9√(5x). dy =

Explanation:

Step1: Recall derivative rules

We know that if $y = ax^n$, then $y^\prime=anx^{n - 1}$, and for a constant - multiple and sum - difference of functions, $(u\pm v)^\prime=u^\prime\pm v^\prime$. Also, $\sqrt{x}=x^{\frac{1}{2}}$.

Step2: Differentiate $5x^4$

For $y_1 = 5x^4$, using the power rule $y_1^\prime=\frac{d}{dx}(5x^4)=5\times4x^{4 - 1}=20x^3$.

Step3: Rewrite and differentiate $9\sqrt{5x}$

First, rewrite $y_2 = 9\sqrt{5x}=9\sqrt{5}x^{\frac{1}{2}}$. Then, using the power rule, $y_2^\prime=\frac{d}{dx}(9\sqrt{5}x^{\frac{1}{2}})=9\sqrt{5}\times\frac{1}{2}x^{\frac{1}{2}-1}=\frac{9\sqrt{5}}{2}x^{-\frac{1}{2}}$.

Step4: Find $dy$

Since $y = 5x^4-9\sqrt{5x}$, then $dy=(y^\prime)dx$. And $y^\prime=\frac{dy}{dx}=20x^3-\frac{9\sqrt{5}}{2\sqrt{x}}$. So $dy=(20x^3-\frac{9\sqrt{5}}{2\sqrt{x}})dx$.

Answer:

$(20x^3-\frac{9\sqrt{5}}{2\sqrt{x}})dx$