Higher Derivatives
Higher Derivatives
- Higher Derivatives involves finding 3rd and higher derivatives of functions.
\(\(\frac{d}{dx}uv=v \frac{du}{dx}+u \frac{dv}{dx}\)\)
\(\((uv)^{'}=u(v)^{'}+(u)^{'}v\)\)
\(y=\ln(1-x)\), find \(\frac{d^3y}{dx^3}\) when \(x=\frac{1}{2}\)
- \(\frac{dy}{dx}\ln(1-x)=\frac{1}{1-x}\times \frac{dy}{dx}(1-x)\)
- \(=-\frac{1}{1-x}\)
- \(\frac{dy}{dx}\left( -\frac{1}{1-x} \right)=\frac{1}{x-1}\)
- \(\frac{d^2y}{dx^2}=-(x-1)^2\)
- \(\frac{d^3y}{dx^3}=2(x-1)^3\)
- Find when \(x=\frac{1}{2}\)
- \(=2\left( \frac{1}{2}-1 \right)^{-3}\)
- \(=2\left( -\frac{1}{2} \right)^{-3}\)
- \(=2(-8)\)
- \(=-16\)
\(f(x)=e^{x^{2}}\)
a) Show that \(f'(x)=2xf(x)\)
- \(e^{x^2}=e^{2x}\)
- \(\frac{dy}{dx}=2xe^{2x}\)
- \(\frac{dy}{dx}=2xe^{x^2}\)
- \(f'(x)=2xf(x)\)
b) Show that:
i) \(f''(x)=2f(x)+2xf'(x)\)
- Using product rule: \((2\times f(x))+(2x\times f'(x))\)
ii) $$