Find the slope of the tangent line of the function ﻿y left parenthesis x right parenthesis space equals space 4 x to the power of 3 divided by 2 end exponent minus 5 x to the power of 1 divided by 2 end exponent space comma space a t space x space equals space 16

To find the slope of the tangent line to the function \( y(x) = \frac{4x^{3/2}}{2} - 5x^{1/2} \) at \( x = 16 \), we need to follow these steps:

### 1. Differentiate the function with respect to \( x \).
First, simplify the given function:

\[
y(x) = 2x^{3/2} - 5x^{1/2}
\]

Now, differentiate \( y(x) \) to find \( \frac{dy}{dx} \), which gives the slope of the tangent line:

\[
\frac{dy}{dx} = \frac{d}{dx} \left( 2x^{3/2} \right) - \frac{d}{dx} \left( 5x^{1/2} \right)
\]

### 2. Differentiate each term:
- For \( 2x^{3/2} \), use the power rule \( \frac{d}{dx} \left( x^n \right) = nx^{n-1} \):

\[
\frac{d}{dx} \left( 2x^{3/2} \right) = 2 \cdot \frac{3}{2}x^{(3/2) - 1} = 3x^{1/2}
\]

- For \( -5x^{1/2} \), apply the power rule:

\[
\frac{d}{dx} \left( -5x^{1/2} \right) = -5 \cdot \frac{1}{2}x^{(1/2) - 1} = -\frac{5}{2}x^{-1/2}
\]

### 3. Combine the derivatives:
Thus, the derivative of the function is:

\[
\frac{dy}{dx} = 3x^{1/2} - \frac{5}{2}x^{-1/2}
\]

### 4. Evaluate the derivative at \( x = 16 \):
Now, substitute \( x = 16 \) into the derivative:

\[
\frac{dy}{dx} \bigg|_{x = 16} = 3(16)^{1/2} - \frac{5}{2}(16)^{-1/2}
\]

Since \( (16)^{1/2} = 4 \) and \( (16)^{-1/2} = \frac{1}{4} \), this simplifies to:

\[
\frac{dy}{dx} \bigg|_{x = 16} = 3(4) - \frac{5}{2} \left( \frac{1}{4} \right)
\]

\[
= 12 - \frac{5}{8}
\]

\[
= \frac{96}{8} - \frac{5}{8} = \frac{91}{8}
\]

### Final Answer:
The slope of the tangent line at \( x = 16 \) is \( \frac{91}{8} \).

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you differentiate functions with fractional exponents?
2. What are the rules for differentiating powers of \( x \)?
3. How do you evaluate expressions involving fractional powers and reciprocals?
4. What does the derivative represent in the context of a tangent line?
5. How does the power rule apply to more complex functions?

### Tip:
When differentiating fractional powers, always simplify your expression before applying the power rule to avoid errors.

Find the slope of the tangent line of the function y(x) = 4x^(3/2)/2 - 5x^(1/2) at x = 16

This problem demonstrates how to find the slope of the tangent line to the function y(x) = 4x^(3/2)/2 - 5x^(1/2) at x = 16. It covers key calculus concepts such as differentiation, the power rule, and evaluating fractional exponents. Suitable for high school students in Grades 10-12.

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