Question - Derivative of a Function with Rational Expression
Solution:
Given the function: $$ h(x) = \frac{4x^3 - 7x + 8}{x} $$First, simplify $$ h(x) $$ by dividing each term in the numerator by $$ x $$:$$ h(x) = 4x^2 - 7 + \frac{8}{x} $$Now, find the derivative $$ h'(x) $$:$$ h'(x) = \frac{d}{dx}(4x^2) - \frac{d}{dx}(7) + \frac{d}{dx}\left(\frac{8}{x}\right) $$$$ h'(x) = 8x - 0 - 8x^{-2} $$$$ h'(x) = 8x - 8x^{-2} $$Expressing $$ -8x^{-2} $$ as $$ -\frac{8}{x^2} $$, we have:$$ h'(x) = 8x - \frac{8}{x^2} $$
Tutor Solution:
To find the derivative of the given function $$ h(x) = \frac{4x^3 - 7x + 8}{x} $$, we need to use the quotient rule which states that if $$ f(x) = \frac{g(x)}{h(x)} $$, then its derivative $$ f'(x) $$ is\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \]However, in this case, it is more straightforward to simplify the function before taking the derivative by dividing each term by $$ x $$. The function simplifies to:\[ h(x) = 4x^2 - 7 + \frac{8}{x} \]Now we can find the derivative by taking the derivative of each term separately. For the terms $$ 4x^2 $$ and $$ -7 $$, we use the power rule, which is $$ \frac{d}{dx}x^n = nx^{n-1} $$. Since $$ -7 $$ is a constant, its derivative is 0.The derivative of $$ \frac{8}{x} $$ is found by rewriting it as $$ 8x^{-1} $$ and then using the power rule.Let's differentiate term-by-term:- The derivative of $$ 4x^2 $$ is $$ 2 \cdot 4x^{2-1} = 8x $$.- The derivative of the constant $$ -7 $$ is $$ 0 $$ (since the derivative of any constant is 0).- The derivative of $$ 8x^{-1} $$ is $$ -1 \cdot 8x^{-1-1} = -8x^{-2} $$, which can also be written as $$ -\frac{8}{x^2} $$.Hence the derivative of $$ h(x) $$ is:\[ h'(x) = 8x - 0 - \frac{8}{x^2} = 8x - \frac{8}{x^2} \]Using proper Latex format, the final answer is:\[ h'(x) = 8x - \frac{8}{x^2} \]
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