Questions: Find the derivative of the function. y=8 sqrt(x)+4 x^(1/7) dy/dx=

Solution Steps

To find the derivative of the function \( y = 8 \sqrt{x} + 4 x^{\frac{1}{7}} \), we will use the power rule for differentiation. The power rule states that the derivative of \( x^n \) is \( n x^{n-1} \). We will apply this rule to each term in the function separately.

We start with the function given by \[ y = 8 \sqrt{x} + 4 x^{\frac{1}{7}}. \]

Using the power rule for differentiation, we differentiate each term:

1. For the term \(8 \sqrt{x}\), we rewrite it as \(8 x^{\frac{1}{2}}\). The derivative is: \[ \frac{d}{dx}(8 x^{\frac{1}{2}}) = 8 \cdot \frac{1}{2} x^{-\frac{1}{2}} = 4 x^{-\frac{1}{2}} = \frac{4}{\sqrt{x}}. \]
2. For the term \(4 x^{\frac{1}{7}}\), the derivative is: \[ \frac{d}{dx}(4 x^{\frac{1}{7}}) = 4 \cdot \frac{1}{7} x^{-\frac{6}{7}} = \frac{4}{7} x^{-\frac{6}{7}} = \frac{4}{7 x^{\frac{6}{7}}}. \]

Now, we combine the derivatives of both terms: \[ \frac{dy}{dx} = \frac{4}{\sqrt{x}} + \frac{4}{7 x^{\frac{6}{7}}}. \]

Final Answer