1. Sum Rule

The derivative of a sum (or difference) is the sum (or difference) of the derivatives.

\[ \frac{d}{dx}[u(x) \pm v(x)] = \frac{d}{dx}u(x) \pm \frac{d}{dx}v(x) \]

2. Constant × a Function

A constant just “comes out” of the derivative.

\[ \frac{d}{dx}[c\,u(x)] = c \frac{d}{dx}u(x) \]

Quick Examples

If \( y = \sin x + \sqrt{x} \) then

\[ \frac{dy}{dx} = \cos x + \frac{1}{2\sqrt{x}} \]

If \( y = 3 \ln x \) then

\[ \frac{dy}{dx} = 3 \frac{1}{x} \]

Problem 1

If \( y = 3x^7 + 4 \sin x - 3 \ln x + \sqrt{a} \), find \( \frac{dy}{dx} \).

Given \( y = 3x^7 + 4 \sin x - 3 \ln x + \sqrt{a} \)

⇒ \[ \frac{dy}{dx} = 3(7x^6) + 4\cos x - 3\frac{1}{x} + 0 \]

i.e. \[ \frac{dy}{dx} = 21x^6 + 4\cos x - \frac{3}{x} \]

Note – In the above problem we treat ‘a’ as a constant, since nothing is specified about ‘a’.

Problem 2

If \( y = \sqrt[3]{x} + 3 \tan x - \sin b \), find \( \frac{dy}{dx} \).

Given \( y = \sqrt[3]{x} + 3 \tan x - \sin b \)

Note that all terms are not differentiable.

i.e. \[ y = x^{\frac{1}{3}} + 3 \tan x - \sin b \]

Differentiating w.r.t \( x \),

\[ \frac{dy}{dx} = \frac{1}{3} x^{\frac{1}{3}-1} + 3 \sec^2 x - 0 \]

\[ \frac{dy}{dx} = \frac{1}{3} x^{-\frac{2}{3}} + 3 \sec^2 x \]

\[ \frac{dy}{dx} = \frac{1}{3 x^{\frac{2}{3}}} + 3 \sec^2 x \]