If $y=(1+x)(1+x^{2})(1+x^{4})$,then $\frac{dy}{dx}$ at $x=1$ is

Differentiate the function with respect to $x$: $x^{x^{2}-3}+(x-3)^{x^{2}}$ for $x > 3$.

$\text{If } \frac{d}{dx} \left( \frac{x^2+1}{(x^2+5)(x^2+9)} \right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)} - \frac{1}{g(x)} - \frac{1}{h(x)} \right], \text{ then } 2h(x) - f(x) - g(x) = $

The derivative of $y = x^{\ln x}$ is

Find the derivative of the function given by $f(x)=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})$ and hence find $f^{\prime}(1).$

If $u, v$ and $w$ are functions of $x,$ then show that $\frac{d}{d x}(u \cdot v \cdot w) = \frac{d u}{d x} \cdot v \cdot w + u \cdot \frac{d v}{d x} \cdot w + u \cdot v \cdot \frac{d w}{d x}$ in two ways - first by repeated application of product rule,second by logarithmic differentiation.