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

Find the derivative of the function.
h(x)=3/x^3 - 9/x^9 + 4 sqrt(x)
h'(x)=
Transcript text: Find the derivative of the function. \[ \begin{array}{l} h(x)=\frac{3}{x^{3}}-\frac{9}{x^{9}}+4 \sqrt{x} \\ h^{\prime}(x)= \end{array} \]
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Solution

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Solution Steps

To find the derivative of the function h(x)=3x39x9+4x 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)=3x39x9+4x1/2 h(x) = 3x^{-3} - 9x^{-9} + 4x^{1/2} . Then, differentiate each term separately using the power rule, which states that the derivative of xn x^n is nxn1 nx^{n-1} .

Step 1: Rewrite the Function in Terms of Exponents

The given function is h(x)=3x39x9+4x h(x) = \frac{3}{x^{3}} - \frac{9}{x^{9}} + 4 \sqrt{x} . We rewrite it using exponents: h(x)=3x39x9+4x1/2 h(x) = 3x^{-3} - 9x^{-9} + 4x^{1/2}

Step 2: Apply the Power Rule to Differentiate

To find the derivative h(x) h'(x) , we apply the power rule to each term. The power rule states that the derivative of xn x^n is nxn1 nx^{n-1} .

  • The derivative of 3x3 3x^{-3} is 9x4 -9x^{-4} .
  • The derivative of 9x9 -9x^{-9} is 81x10 81x^{-10} .
  • The derivative of 4x1/2 4x^{1/2} is 2x1/2 2x^{-1/2} .
Step 3: Combine the Derivatives

Combine the derivatives of each term to find h(x) h'(x) : h(x)=9x4+81x10+2x h'(x) = -\frac{9}{x^4} + \frac{81}{x^{10}} + \frac{2}{\sqrt{x}}

Final Answer

The derivative of the function is: h(x)=9x4+81x10+2x \boxed{h'(x) = -\frac{9}{x^4} + \frac{81}{x^{10}} + \frac{2}{\sqrt{x}}}

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