Question

find the derivative: 3. ( y = 4x^3 + 6sqrt{x} - \frac{3}{x^2} + dots ) (partial text visible, including terms like ( ln|x| ), ( 3sin x ) possibly)

Explanation:

Assuming the function is \( y = 4x^{3}+6\sqrt{x}-\frac{3}{x^{2}} \) (correcting possible typos), we find its derivative using power rule.

Step1: Differentiate \( 4x^{3} \)

Power rule: \( \frac{d}{dx}(x^{n}) = nx^{n - 1} \). For \( 4x^{3} \), derivative is \( 4\times3x^{3 - 1}=12x^{2} \).

Step2: Differentiate \( 6\sqrt{x}=6x^{\frac{1}{2}} \)

Using power rule: \( 6\times\frac{1}{2}x^{\frac{1}{2}-1}=3x^{-\frac{1}{2}}=\frac{3}{\sqrt{x}} \).

Step3: Differentiate \( -\frac{3}{x^{2}}=-3x^{-2} \)

Power rule: \( -3\times(-2)x^{-2 - 1}=6x^{-3}=\frac{6}{x^{3}} \).

Step4: Combine derivatives

Sum the derivatives: \( y' = 12x^{2}+\frac{3}{\sqrt{x}}+\frac{6}{x^{3}} \).

Answer:

\( y' = 12x^{2}+\frac{3}{\sqrt{x}}+\frac{6}{x^{3}} \)