Derivative of a Function with Simplification
The function \( h(x) \) is given as:
\[ h(x) = \frac{4x^3 - 7x + 8}{x} \]
Before we find the derivative, we can simplify the function by dividing each term in the numerator by \( x \). This gives us:
\[ h(x) = 4x^2 - 7 + \frac{8}{x} \]
Now, let's find the derivative of \( h(x) \) with respect to \( x \) using the power rule and the derivative of a constant. The power rule states that the derivative of \( x^n \) with respect to \( x \) is \( n \cdot x^{n-1} \), and the derivative of a constant is 0. For the term \( \frac{8}{x} \), which can be written as \( 8x^{-1} \), we also use the power rule.
Let's differentiate each term:
1. The derivative of \( 4x^2 \) with respect to \( x \) is \( 2 \cdot 4x = 8x \).
2. The derivative of \( -7 \) (a constant) with respect to \( x \) is \( 0 \).
3. The derivative of \( \frac{8}{x} \) (which is \( 8x^{-1} \)) with respect to \( x \) is \( -1 \cdot 8x^{-1 - 1} = -8x^{-2} \).
Putting it all together, the derivative of \( h(x) \) is:
\[ h'(x) = 8x - 0 - \frac{8}{x^2} \]
\[ h'(x) = 8x - \frac{8}{x^2} \]
So, the derivative of the function \( h(x) \) is \( 8x - \frac{8}{x^2} \).
