Question - Differentiation of a Rational Function
Solution:
The function provided in the image is:\[ y = \frac{x^5 - 5x^6 + 5}{x^4} \]To differentiate this function with respect to $$ x $$, you can first simplify the expression by dividing each term by $$ x^4 $$. This gives us:\[ y = x - 5x^2 + \frac{5}{x^4} \]Now, let's differentiate term by term using basic differentiation rules:1. The derivative of $$ x $$ with respect to $$ x $$ is 1.2. The derivative of $$ -5x^2 $$ with respect to $$ x $$ is $$ -10x $$.3. The derivative of $$ \frac{5}{x^4} $$ with respect to $$ x $$ is $$ -20x^{-5} $$ (using the power rule and multiplying by the constant 5).Combining these results:\[ \frac{dy}{dx} = 1 - 10x - 20x^{-5} \]Or, if we express the last term with positive exponents:\[ \frac{dy}{dx} = 1 - 10x - \frac{20}{x^5} \]
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