5 calculus
5.1 continuity and differentiability
\(f(x)\) is continous at \(x=a\) where
- \(f(a)\) is defined
- \(\lim_{x\to{}a}f(x)=f(a)\)
a function is differentiable if it is continuous and its derivative is also continuous at all values of \(x\)
5.2 differentiation
if \(f'(x)\) is the derivative function of a function \(f(x)\) then \(f'(a)\) returns the gradient of the function \(f(x)\) at \(x=a\)
\[f'(x)=\lim_{h\to{}0}\frac{f(x+h)-f(x)}{h}\]
5.2.1 common derivative
| \(f(x)\) | \(f'(x)\) |
|---|---|
| \(x^n\) | \(nx^{n-1}\) |
| \((ax+b)^n\) | \(an(ax+b)^{n-1}\) |
| \(e^{f(x)}\) | \(f'(x)e^{f(x)}\) |
| \(\log_ef(x)\) | \(\frac{f'(x)}{f(x)}\) |
| \(\sin{f(x)}\) | \(f'(x)\cos{f(x)}\) |
| \(\cos{f(x)}\) | \(-f'(x)\sin{f(x)}\) |
| \(\tan{f(x)}\) | \(f'(x)\sec^2{f(x)}\) |
5.2.2 chain rule
\[\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\]
5.2.3 product rule
\[\frac{d}{dx}uv=v\frac{du}{dx}+u\frac{dv}{dx}\]
5.2.4 quotient rule
\[\frac{d}{dx}uv=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\]