26. Given that \( y=e^{m \tan ^{-1} x^{2}} \), show that \( \left(1+x^{4}\right) \frac{d y}{d x}=2 m x y \)

To show that \((1 + x^{4}) \frac{dy}{dx} = 2mx y\) given \( y = e^{m \tan^{-1} x^{2}} \), follow these steps: ### Step 1: Differentiate \( y \) with respect to \( x \) Given: \[ y = e^{m \tan^{-1} x^{2}} \] Using the chain rule: \[ \frac{dy}{dx} = y \cdot \frac{d}{dx}\left(m \tan^{-1} x^{2}\right) \] \[ \frac{dy}{dx} = y \cdot m \cdot \frac{d}{dx}\left(\tan^{-1} x^{2}\right) \] ### Step 2: Differentiate \( \tan^{-1} x^{2} \) with respect to \( x \) Recall that: \[ \frac{d}{dx} \tan^{-1} u = \frac{1}{1 + u^{2}} \cdot \frac{du}{dx} \] Let \( u = x^{2} \), then: \[ \frac{d}{dx} \tan^{-1} x^{2} = \frac{1}{1 + (x^{2})^{2}} \cdot 2x = \frac{2x}{1 + x^{4}} \] ### Step 3: Substitute back into the derivative of \( y \) Plugging this result back: \[ \frac{dy}{dx} = y \cdot m \cdot \frac{2x}{1 + x^{4}} = \frac{2mx y}{1 + x^{4}} \] ### Step 4: Multiply both sides by \( 1 + x^{4} \) To obtain the desired equation: \[ (1 + x^{4}) \frac{dy}{dx} = 2mx y \] ### Conclusion Thus, we have successfully shown that: \[ (1 + x^{4}) \frac{dy}{dx} = 2mx y \] **Answer:** After expanding, the derivative is (2mx y)/(1 + x⁴). Thus, (1 + x⁴) dy/dx = 2m x y.