Leibniz's Rule For nth Derivative of a Product
\[uv\]
is\[\frac{d}{dx}(uv) =u \frac{d^nv}{dx}+{}^nC_1 \frac{du}{dx} \frac{d^{n-1}v}{dx^{n-1}}+ {}^nC_2 \frac{d^2u}{dx^2} \frac{d^{n-2}v}{dx^{n-2}}+ ...+ \frac{d^nu}{dx^n} v \]
.Example:
\[f(x)=x^4e^x\]
.\[\begin{equation} \begin{aligned} \frac{d^5}{dx^5}(x^4e^x) &= x^4 \frac{d^5(e^x)}{dx^5}+ {}^5C_1 \frac{d(x^4)}{dx} \frac{d^4(e^x)}{dx^4}+ {}^5C_2 \frac{d^2(x^4)}{dx^2} \frac{d^3(e^x)}{dx^3} \\ &+ {}^5C_3 \frac{d^3(x^4)}{dx^3} \frac{d^2(e^x)}{dx^2}+ {}^5C_4 \frac{d^4(x^4)}{dx^4} \frac{d(e^x)}{dx}++ \frac{d^5(x^4)}{dx^5} e^x \\ &= x^4e^x+20x^3e^x+120x^2e^x+240xe^x+120e^x \end{aligned} \end{equation}\]
