Mathematics

To differentiate the function \( h(x) = \frac{4x^3 – 7x + 8}{x} \) ,here’s the step-by-step process:

Given:

\[
h(x) = \frac{4x^3 – 7x + 8}{x}
\]

Step 1: Simplify the function
First, simplify the function by dividing each term in the numerator by \( x \):

\[
h(x) = \frac{4x^3}{x} – \frac{7x}{x} + \frac{8}{x}
\]

This simplifies to:

\[
h(x) = 4x^2 – 7 + \frac{8}{x}
\]

Step 2: Differentiate each term
Now, differentiate \( h(x) \) with respect to \( x \):

1. Differentiate \( 4x^2 \):
\[
\frac{d}{dx}(4x^2) = 8x
\]

2. Differentiate \( -7 \)(a constant):
\[
\frac{d}{dx}(-7) = 0
\]

3. Differentiate \( \frac{8}{x} \):
Rewrite \( \frac{8}{x} \) as \( 8x^{-1} \).
\[
\frac{d}{dx}(8x^{-1}) = -8x^{-2}
\]

Step 3: Combine the derivatives
Finally, combine the derivatives:

\[
h'(x) = 8x + 0 – \frac{8}{x^2}
\]

Or, simply:

\[
h'(x) = 8x – \frac{8}{x^2}
\]

This is the derivative of the given function \( h(x) = \frac{4x^3 – 7x + 8}{x} \).