Example Question - basic differentiation rules

Differentiation of a Rational Function

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: \[ \frac{dy}{dx} = 1 - 10x - \frac{20}{x^5} \]