Questions: Find the derivative of the following function. y=5x^(-3/2)+4x^(-1/2)+x^5-4 y'=

Transcript text: Find the derivative of the following function. \[ \begin{array}{l} y=5 x^{-\frac{3}{2}}+4 x^{-\frac{1}{2}}+x^{5}-4 \\ y^{\prime}=\square \end{array} \]

Solution

Solution Steps

Step 1: Define the Function

Given the function: \[ y = 5x^{-\frac{3}{2}} + 4x^{-\frac{1}{2}} + x^5 - 4 \]

Step 2: Apply the Power Rule

To find the derivative \( y' \), apply the power rule to each term separately: \[ \frac{d}{dx} \left( 5x^{-\frac{3}{2}} \right) = 5 \cdot -\frac{3}{2} x^{-\frac{3}{2} - 1} = -\frac{15}{2} x^{-\frac{5}{2}} \] \[ \frac{d}{dx} \left( 4x^{-\frac{1}{2}} \right) = 4 \cdot -\frac{1}{2} x^{-\frac{1}{2} - 1} = -2 x^{-\frac{3}{2}} \] \[ \frac{d}{dx} \left( x^5 \right) = 5x^{4} \] \[ \frac{d}{dx} \left( -4 \right) = 0 \]

Step 3: Combine the Derivatives

Combine the derivatives of the individual terms to obtain the derivative of the entire function: \[ y' = -\frac{15}{2} x^{-\frac{5}{2}} - 2 x^{-\frac{3}{2}} + 5x^4 \]