Derivative of a Function with a Rational Expression
To find the derivative of the function \( h(x) \) given by:
\[ h(x) = \frac{4x^3 - 7x + 8}{x} \]
you can simplify the function before differentiating it by dividing each term in the numerator by \( x \):
\[ h(x) = \frac{4x^3}{x} - \frac{7x}{x} + \frac{8}{x} \]
\[ h(x) = 4x^2 - 7 + \frac{8}{x} \]
Now, we'll differentiate each term separately:
- The derivative of \( 4x^2 \) with respect to \( x \) is \( 8x \).
- The derivative of a constant, \( -7 \), is 0.
- The derivative of \( \frac{8}{x} \), which is \( 8x^{-1} \), is \( -8x^{-2} \) or \( -\frac{8}{x^2} \).
So, the derivative of \( h(x) \), which we denote as \( h'(x) \), is:
\[ h'(x) = 8x - 0 - \frac{8}{x^2} \]
\[ h'(x) = 8x - \frac{8}{x^2} \]
This is the final form of the derivative of \( h(x) \).
