Q.a) Find the derivative of $y = \sin x$ from the first principle.

Quotient Rule Differentiation

Suppose a quantity is a ratio $\dfrac{u}{v}$ where both the top $u$ and the bottom $v$ change with $x$. How fast does the ratio itself change? Naively dividing the derivative of the top by the derivative of the bottom does not work — the quotient rule tells you the correct answer.

The Intuition

If the numerator grows, the fraction grows; if the denominator grows, the fraction shrinks. The quotient rule balances these two competing effects:

The derivative of a quotient is (bottom times derivative of top) minus (top times derivative of bottom), all divided by bottom squared.

Here $u = u(x)$ and $v = v(x)$ are functions of $x$, and $u' = \dfrac{du}{dx}$, $v' = \dfrac{dv}{dx}$.

A Way to Remember It

A common mnemonic is "low d-high minus high d-low, over low squared":

• "low" $= v$ (the denominator),
• "d-high" $= u'$ (derivative of the numerator),
• "high" $= u$ (the numerator),
• "d-low" $= v'$ (derivative of the denominator).

Worked Example

Differentiate $f(x) = \dfrac{x^2 + 1}{x - 2}$.

Take $u = x^2 + 1$ and $v = x - 2$, so $u' = 2x$ and $v' = 1$.

Expand and simplify the numerator:

When to Use It

Reach for the quotient rule only when one function is genuinely divided by another and you cannot simplify. For instance, $\dfrac{x^2 + 1}{x}$ is better rewritten as $x + \dfrac{1}{x}$ and differentiated term by term, while $\dfrac{x^2 + 1}{x - 2}$ has no such simplification. …