Find the first, second, and third derivatives of the function \( h(x)=7 \sqrt{x} \)

To find the first, second, and third derivatives of the function \( h(x) = 7 \sqrt{x} \), we can start by rewriting the function in a more derivative-friendly form: \[ h(x) = 7 x^{1/2} \] Now let's calculate the derivatives: 1. **First Derivative**: Using the power rule, \( \frac{d}{dx} x^n = n x^{n-1} \): \[ h'(x) = 7 \cdot \frac{1}{2} x^{1/2 - 1} = \frac{7}{2} x^{-1/2} = \frac{7}{2 \sqrt{x}} \] 2. **Second Derivative**: We apply the power rule again on the first derivative: \[ h''(x) = \frac{d}{dx} \left( \frac{7}{2} x^{-1/2} \right) = \frac{7}{2} \cdot \left(-\frac{1}{2} x^{-3/2}\right) = -\frac{7}{4} x^{-3/2} = -\frac{7}{4 \sqrt{x^3}} \] 3. **Third Derivative**: Now we differentiate the second derivative: \[ h'''(x) = \frac{d}{dx} \left(-\frac{7}{4} x^{-3/2}\right) = -\frac{7}{4} \cdot \left(-\frac{3}{2} x^{-5/2}\right) = \frac{21}{8} x^{-5/2} = \frac{21}{8 \sqrt{x^5}} \] In summary: - The first derivative is \( h'(x) = \frac{7}{2 \sqrt{x}} \). - The second derivative is \( h''(x) = -\frac{7}{4 \sqrt{x^3}} \). - The third derivative is \( h'''(x) = \frac{21}{8 \sqrt{x^5}} \).