Question - Derivative of a Function with a Rational Expression
mathderivative of functionrational expressiondifferentiating rational functionssimplifying functions
Solution:
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 $$ h(x) $$.
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