Questions: Find the derivative of the function. h(x)=3/x^3 - 9/x^9 + 4 sqrt(x) h'(x)=

Solution Steps

To find the derivative of the function \( h(x) = \frac{3}{x^{3}} - \frac{9}{x^{9}} + 4 \sqrt{x} \), we will apply the power rule and the constant multiple rule. First, rewrite the function in terms of exponents: \( h(x) = 3x^{-3} - 9x^{-9} + 4x^{1/2} \). Then, differentiate each term separately using the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \).

The given function is \( h(x) = \frac{3}{x^{3}} - \frac{9}{x^{9}} + 4 \sqrt{x} \). We rewrite it using exponents: \[ h(x) = 3x^{-3} - 9x^{-9} + 4x^{1/2} \]

To find the derivative \( h'(x) \), we apply the power rule to each term. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).

• The derivative of \( 3x^{-3} \) is \( -9x^{-4} \).
• The derivative of \( -9x^{-9} \) is \( 81x^{-10} \).
• The derivative of \( 4x^{1/2} \) is \( 2x^{-1/2} \).

Combine the derivatives of each term to find \( h'(x) \): \[ h'(x) = -\frac{9}{x^4} + \frac{81}{x^{10}} + \frac{2}{\sqrt{x}} \]

Final Answer