Q.Using the rule for differentiation for quotient of two functions, prove that $\dfrac{d}{dx}\left(\dfrac{\sin x}{\cos x}\right) = \sec^2 x$

Differentiation finds the instantaneous RATE OF CHANGE of one quantity with respect to another. For a function $y=f(x)$, the average rate of change between two nearby points is the slope of the chord joining them, $\Delta y/\Delta x$. As the separation $\Delta x$ between the points shrinks to zero, the chord becomes the TANGENT to the curve at that point, and the limiting value of the ratio $\Delta y/\Delta x$ — which need not itself go to zero even though both $\Delta x$ and $\Delta y$ do — is called the derivative of $y$ with respect to $x$, written $dy/dx$: $\dfrac{dy}{dx} = \lim_{\Delta x\to0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$.

A handful of standard rules make differentiation of combined functions manageable: the constant-multiple rule ($\frac{d}{dx}(sf)=s\frac{df}{dx}$), the sum rule (differentiate term-by-term), the product rule ($\frac{d}{dx}(f_1f_2)=f_1\frac{df_2}{dx}+f_2\frac{df_1}{dx}$), the quotient rule, and the chain rule (for a function of a function, or when the variable itself depends on a further variable such as time). …