Questions: Differentiate the function. g(x)=1/(sqrt(x))+sqrt[5](x) g'(x)=

Transcript text: Differentiate the function. \[ \begin{array}{r} g(x)=\frac{1}{\sqrt{x}}+\sqrt[5]{x} \\ g^{\prime}(x)=\square \end{array} \]

Solution

Solution Steps

To differentiate the function \( g(x) = \frac{1}{\sqrt{x}} + \sqrt[5]{x} \), we will apply the rules of differentiation. The first term can be rewritten as \( x^{-1/2} \) and the second term as \( x^{1/5} \). We will use the power rule for differentiation, which states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \).

Step 1: Rewrite the Function

The given function is \( g(x) = \frac{1}{\sqrt{x}} + \sqrt[5]{x} \). We can rewrite this function using exponents: \[ g(x) = x^{-1/2} + x^{1/5} \]

Step 2: Apply the Power Rule

To differentiate the function, we apply the power rule, which states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \).

For the first term \( x^{-1/2} \), the derivative is: \[ -\frac{1}{2} \cdot x^{-3/2} = -\frac{1}{2x^{3/2}} \]
For the second term \( x^{1/5} \), the derivative is: \[ \frac{1}{5} \cdot x^{-4/5} = \frac{1}{5x^{4/5}} \]

Step 3: Combine the Derivatives

Combine the derivatives of the two terms to find \( g'(x) \): \[ g'(x) = -\frac{1}{2x^{3/2}} + \frac{1}{5x^{4/5}} \]