$y=x(x+2)^{3}$ Let $u=x$ and $v=(x+2)^{3}$ $\dfrac{du}{dx}=1$ and $\dfrac{dv}{dx}=3(x+2)^{2}$ $\dfrac{dy}{dx}=v\dfrac{du}{dx}+u\dfrac{dv}{dx}$ $\dfrac{dy}{dx}=(x+2)^{3}\times 1+x\times 3(x+2)^{2}$ $\dfrac{dy}{dx}=(x+2)^{3}+3x(x+2)^{2}$ $\dfrac{dy}{dx}=(x+2)^{2}(x+2+3x)$ $\dfrac{dy}{dx}=(x+2)^{2}(4x+Read more

$y=(x+1)^{3}$ Let $u=x+1$ and $y=u^{3}$ $\dfrac{du}{dx}=1$ and $\dfrac{dy}{du}=3u^{2}$ $\dfrac{dy}{dx}=\dfrac{du}{dx}\times \dfrac{dy}{du}$ $\dfrac{dy}{dx}=1\times 3u^{2}$ $\dfrac{dy}{dx}=3xu^{2}=3x(x+1)^{2}$ $\dfrac{d^{2}y}{dx^{2}}=(2x\times 2)\times 3x(x^{2}-1)^{2-1}$ $\dfrac{d^{2}y}{dx^{2}}=12x^{Read more

$y=(x^{2}-1)^{3}$ Let $u=x^{2}-1$ and $y=u^{3}$ $\dfrac{du}{dx}=2x$ and $\dfrac{dy}{du}=3u^{2}$ $\dfrac{dy}{dx}=\dfrac{du}{dx}\times \dfrac{dy}{du}$ $\dfrac{dy}{dx}=2x\times 3u^{2}$ $\dfrac{dy}{dx}=6xu^{2}=6x(x^{2}-1)^{2}$ $\dfrac{d^{2}y}{dx^{2}}=(2x\times 2)\times 6x(x^{2}-1)^{2-1}$ $\dfrac{d^{2}y}Read more

$y=x^{-1}-x^{-2}$ $\dfrac{dy}{dx}=-1\times x^{-1-1}-(-2)\times x^{-2-1}$ $\dfrac{dy}{dx}=-x^{-2}+2x^{-3}$ $\dfrac{dy}{dx}=-\dfrac{1}{x^{2}}+\dfrac{1}{x^{3}}$ $\dfrac{d^{2}y}{dx^{2}}=-2\times -x^{-2-1}+(-3)\times x^{-3-1}$ $\dfrac{d^{2}y}{dx^{2}}=2x^{-3}-3x^{-4}$ $\dfrac{d^{2}y}{dx^{2}}=}=\dfrac{2}{xRead more

$y=3x^{4}-2x^{2}-x^{-1}$ $\dfrac{dy}{dx}=4\times 3x^{4-1}-2\times 2x^{2-1}-(-1)\times x^{-1-1}$ $\dfrac{dy}{dx}=12x^{3}-4x+x^{-2}$ $\dfrac{dy}{dx}=12x^{3}-4x+\dfrac{1}{x^{2}}$ $\dfrac{d^{2}y}{dx^{2}}=3\times 12x^{3-1}-4x^{1-1}+(-2)\times x^{-2-1}$ $\dfrac{d^{2}y}{dx^{2}}=36x^{2}-4-2x^{-3}$ $\dfrac{dRead more