Standard Derivatives Of Some Basic Functions - SS3 Mathematics Lesson Note
|
\[\mathbf{x}\] |
\[\mathbf{f'(x)}\]
|
|---|---|
|
\[a\] |
\[0\] |
|
\[{ax}^{n}\] |
\[{anx}^{n - 1}\] |
|
\[\sin x\] |
\[\cos x\] |
|
\[\cos x\] |
\[- \sin x\] |
|
\[\tan x\] |
\[\sec^{2}x\] |
|
\[e^{x}\] |
\[e^{x}\] |
|
\[a^{x}\] |
\[a^{x}\log_{e}a\] |
|
\[\log_{e}x\] |
\[\frac{1}{x}\] |
|
\[\log_{a}x\] |
\[\frac{1}{x\ \log_{e}a}\] |
|
\[\cot x\] |
\[- {cosec}^{2}x\ \] |
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) \(y = 2\log_{10}x\)
\[\frac{dy}{dx} = \frac{d}{dx}(2\log_{10}x)\]
\[\frac{dy}{dx} = 2\frac{d}{dx}(\log_{10}x)\]
\[\frac{dy}{dx} = 2\left( \frac{1}{x\ \log_{e}10} \right)\]
\[\frac{dy}{dx} = \frac{2}{x\ \log_{e}10}\]