Differentiation – Rules
Product Rule of Differentiation: The derivative of the product of two functions
\(\frac{d}{dx}\left\{ f\left( x \right).g\left( x \right) \right\}=f\left( x \right).\frac{d}{dx}\left\{ g\left( x \right) \right\}+g\left( x \right).\frac{d}{dx}\left\{ f\left( x \right) \right\}\).
= (First Function) x (Derivative of Second Function) + (Second Function) x (Derivative of First Function)
Quotient Rule of Differentiation: The derivative of the quotient of two functions
\(\frac{d}{dx}\left\{ \frac{f\left( x \right)}{g\left( x \right)} \right\}=\frac{g\left( x \right).\frac{d}{dx}\left\{ f\left( x \right) \right\}-f\left( x \right).\frac{d}{dx}\left\{ g\left( x \right) \right\}}{{{\left\{ g\left( x \right) \right\}}^{2}}}\).
\(=\frac{(Deno\min ator\,\,\times Derivative\,\,of\,\,Numerator)-(Numerator\times Derivative\,\,of\,\,Deno\min ator)}{{{(Deno\min ator)}^{2}}}\).
Derivative of a Function: (Chain Rule) If y is a differentiable function of t and t is a differentiable function of x, i.e., y = f(t) and t = g(x), then
\(\frac{dy}{dx}=\frac{dy}{dt}.\frac{dt}{dx}\)Similarly, if y = f(u), where u = g(v) and v = h(x), then
\(\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dv}.\frac{dv}{dx}\).
Derivative of Parametric Functions: Sometimes x and y are separately given as functions of a single variable t (called a parameter), i.e., x = f(t) and y = g(t). In this case,
\(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{f'(t)}{g'(t)}\).
And \(\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{d}{dx}\left( \frac{dy}{dx} \right)\)
\(\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{d}{dt}\left( \frac{dy}{dx} \right)\times \frac{dt}{dv}\).