Questions: Find the derivative of (f(x)=2 sqrtx+frac4x^3).

$Find the derivative of (f(x)=2 sqrtx+frac4x^3).$

Transcript text: Find the derivative of $f(x)=2 \sqrt{x}+\frac{4}{x^{3}}$. Type your answer without fractional or negative exponents. \[ f^{\prime}(x)= \] $\square$

Solution

Solution Steps

To find the derivative of the function $ f(x) = 2 \sqrt{x} + \frac{4}{x^3} $, we will apply the rules of differentiation. The function consists of two terms: $ 2 \sqrt{x} $ and $ \frac{4}{x^3} $. We will differentiate each term separately. For the first term, $ 2 \sqrt{x} $, we can rewrite it as $ 2x^{1/2} $ and use the power rule. For the second term, $ \frac{4}{x^3} $, we can rewrite it as $ 4x^{-3} $ and again use the power rule. Finally, we will combine the derivatives of both terms to get the derivative of the entire function.

Step 1: Rewrite the Function for Differentiation

The function given is $ f(x) = 2 \sqrt{x} + \frac{4}{x^3} $. We can rewrite this function using exponents to make differentiation easier: \[ f(x) = 2x^{1/2} + 4x^{-3} \]

Step 2: Differentiate Each Term

To find the derivative $ f'(x) $, we differentiate each term separately using the power rule:

For $ 2x^{1/2} $, the derivative is: \[ \frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{1/2 - 1} = x^{-1/2} = \frac{1}{\sqrt{x}} \]
For $ 4x^{-3} $, the derivative is: \[ \frac{d}{dx}(4x^{-3}) = 4 \cdot (-3)x^{-3 - 1} = -12x^{-4} = -\frac{12}{x^4} \]

Step 3: Combine the Derivatives

Combine the derivatives of the individual terms to find the derivative of the entire function: \[ f'(x) = \frac{1}{\sqrt{x}} - \frac{12}{x^4} \]