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