Standard Derivatives Of Some Basic Functions - SS3 Mathematics Lesson Note

\[\mathbf{f'(x)}\]

 

\[a\]

\[0\]

\[{ax}^{n}\]

\[{anx}^{n - 1}\]

\[\sin x\]

\[\cos x\]

\[\cos x\]

\[- \sin x\]

\[\tan x\]

\[\sec^{2}x\]

\[e^{x}\]

\[e^{x}\]

\[a^{x}\]

\[a^{x}\log_{e}a\]

\[\log_{e}x\]

\[\frac{1}{x}\]

\[\log_{a}x\]

\[\frac{1}{x\ \log_{e}a}\]

\[\cot x\]

\[- {cosec}^{2}x\ \]

Where \(a\) is a constant

Example 2 Obtain the derivative of the following functions of \(x\): (i) \(y = 2x^{2} - 7\) (ii) \(y = 3e^{x}\) (iii) \(y = - 2\cos x\) (iv) \(y = 3x^{- 4}\) (v) \(y = 2^{x}\) (vi) \(y = 2\log_{10}x\)

Solution

(i) \(y = 2x^{2} - 7\)

\[\frac{dy}{dx} = \frac{d}{dx}\left( 2x^{2} \right) - \frac{d}{dx}(7)\]

\[\frac{dy}{dx} = 2(2x^{2 - 1}) - 0\]

\[\frac{dy}{dx} = 2(2x^{1}) - 0\]

\[\frac{dy}{dx} = 4x - 0\]

\[\frac{dy}{dx} = 4x\]

(ii) \(y = 3e^{x}\)

\[\frac{dy}{dx} = 3\frac{d}{dx}(e^{x})\]

\[\frac{dy}{dx} = 3(e^{x})\]

\[\frac{dy}{dx} = 3e^{x}\]

(iii) \(y = - 2\cos x\)

\[\frac{dy}{dx} = - 2\frac{d}{dx}(\cos x)\]

\[\frac{dy}{dx} = - 2( - \sin x)\]

\[\frac{dy}{dx} = 2\sin x\]

(iv) \(y = 3x^{- 4}\)

\[\frac{dy}{dx} = 3\frac{d}{dx}(x^{- 4})\]

\[\frac{dy}{dx} = 3({- 4x}^{- 4 - 1})\]

\[\frac{dy}{dx} = 3({- 4x}^{- 5})\]

\[\frac{dy}{dx} = {- 12x}^{- 5}\]

(v) \(y = 2^{x}\)

\[\frac{dy}{dx} = \frac{d}{dx}(2^{x})\]

\[\frac{dy}{dx} = 2^{x}\log_{e}2\]

(vi) \(y = 2\log_{10}x\)

\[\frac{dy}{dx} = \frac{d}{dx}(2\log_{10}x)\]

\[\frac{dy}{dx} = 2\frac{d}{dx}(\log_{10}x)\]

\[\frac{dy}{dx} = 2\left( \frac{1}{x\ \log_{e}10} \right)\]

\[\frac{dy}{dx} = \frac{2}{x\ \log_{e}10}\]