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NSW Y12 Maths - Advanced Differential Calculus Second Derivatives

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Second Derivatives Theory

For \(y=f(x)\) the first derivative is \(\dfrac{d y}{d x}\) or \(f^{\prime}(x)\). \\ The second derivative results in the derivative of the first derivative lie. \(\dfrac{d^2 y}{d x^2}\) or \(f^{\prime \prime}(x)\).\\ \textbf{Example}\\ %16699 The second derivative of \(\dfrac{1}{{\sqrt {1 + {x^2}} }}\) is?\\ \textbf{Solution}\\ $\begin{aligned} &y=\frac{1}{\sqrt{1+x^{2}}}\\ &y=\left(1+x^{2}\right)^{-\frac{1}{2}} \end{aligned}$\\ $\begin{aligned} y^{\prime} &=-\frac{1}{2}\left(1+x^{2}\right)^{-\frac{3}{2}} \times 2 x \\ &=-x\left(1+x^{2}\right)^{-\frac{3}{2}}\\ \text { let } u &=-x \\ u^{\prime} &=-1\\ v &=\left(1+x^{2}\right)^{-\frac{-3}{2}} \\ v^{\prime} &=-\frac{3}{2}\left(1+x^{2}\right)^{-\frac{5}{2}} 2 x \\ &=-3 x\left(1+x^{2}\right)^{-\frac{5}{2}} \\ y^{\prime}&=uv^{l}+v u^{\prime}\\ &=3 x^{2}\left(1+x^{2}\right)^{-\frac{5}{2}}-\left(1+x^{2}\right)^{-\frac{3}{2}}\\ &=\left(1+x^{2}\right)^{-\frac{5}{2}}\left(3 x^{2}-\left(1+x^{2}\right)^{1}\right)\\ &=\frac{2 x^{2}-1}{\sqrt{\left(1+x^{2}\right)^{5}}} \end{aligned}$\\

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  • Second Derivatives - Video - Finding first and second derivative

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