Questions: 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.
\[
\begin{array}{l}
h(x)=\frac{3}{x^{3}}-\frac{9}{x^{9}}+4 \sqrt{x} \\
h^{\prime}(x)=
\end{array}
\]
Solution
Solution Steps
To find the derivative of the function h(x)=x33−x99+4x, 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+4x1/2. Then, differentiate each term separately using the power rule, which states that the derivative of xn is nxn−1.
Step 1: Rewrite the Function in Terms of Exponents
The given function is h(x)=x33−x99+4x. We rewrite it using exponents:
h(x)=3x−3−9x−9+4x1/2
Step 2: Apply the Power Rule to Differentiate
To find the derivative h′(x), we apply the power rule to each term. The power rule states that the derivative of xn is nxn−1.
The derivative of 3x−3 is −9x−4.
The derivative of −9x−9 is 81x−10.
The derivative of 4x1/2 is 2x−1/2.
Step 3: Combine the Derivatives
Combine the derivatives of each term to find h′(x):
h′(x)=−x49+x1081+x2
Final Answer
The derivative of the function is:
h′(x)=−x49+x1081+x2