Question - Derivative of a Function with Simplification
mathpower rule differentiationderivative of functionderivative simplificationfunction derivative example
Solution:
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 $$ 8x - \frac{8}{x^2} $$.
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