Questions: Question 20 Use logarithmic differentiation to find the derivative of the function f(x) = x^3 / (x^4 + 1)^2

Solution Steps

We start with the function given by \[ f(x) = \frac{x^{3}}{(x^{4} + 1)^{2}}. \]

Taking the natural logarithm of both sides, we have: \[ \ln(f) = \ln\left(\frac{x^{3}}{(x^{4} + 1)^{2}}\right) = \ln(x^{3}) - \ln((x^{4} + 1)^{2}) = 3\ln(x) - 2\ln(x^{4} + 1). \]

Differentiating both sides with respect to \( x \): \[ \frac{d}{dx}(\ln(f)) = \frac{1}{f} \frac{df}{dx} = \frac{3}{x} - \frac{8x^{3}}{x^{4} + 1}. \]

Multiplying both sides by \( f \): \[ \frac{df}{dx} = f \left( \frac{3}{x} - \frac{8x^{3}}{x^{4} + 1} \right). \] Substituting back \( f(x) \): \[ \frac{df}{dx} = \frac{x^{3}}{(x^{4} + 1)^{2}} \left( \frac{3}{x} - \frac{8x^{3}}{x^{4} + 1} \right). \]

After simplification, we find: \[ \frac{df}{dx} = \frac{x^{2}(3 - 5x^{4})}{(x^{4} + 1)^{3}}. \]

Final Answer