Differentiation

Differentiation Rules
1.\,\frac{\text{d}}{\text{d}x} c = 0
2.\,\frac{\text{d}}{\text{d}x} x^n = nx^{n-1}
3.\,\frac{\text{d}}{\text{d}x} \sin x = \cos  x
4.\,\frac{\text{d}}{\text{d}x} \cos x = -\sin  x
5.\,\frac{\text{d}}{\text{d}x} \tan x = \sec^2 x
6.\,\frac{\text{d}}{\text{d}x} e^{x} = e^{x}
7.\,\frac{\text{d}}{\text{d}x} \ln\; x = \frac{1}{x}
Note: \frac{\text{d}}{\text{d}x} k\text{f}(x)=k\times\frac{\text{d}}{\text{d}x} \text{f}(x)
Chain Rule
For y=\text{f}(u) and u=\text{g}(x),
\dfrac{\text{d}y}{\text{d}x} = \dfrac{\text{d}y}{\text{d}u} \times \dfrac{\text{d}u}{\text{d}x}
Further Differentiation Rules (Chain Rule)
1.\,\frac{\text{d}}{\text{d}x} (ax+b)^n = an(ax+b)^{n-1}
2.\,\frac{\text{d}}{\text{d}x} \sin (ax+b) = a\cos  (ax+b)
3.\,\frac{\text{d}}{\text{d}x} \cos (ax+b) = -a\sin  (ax+b)
4.\,\frac{\text{d}}{\text{d}x} \tan (ax+b) = a\sec^2 (ax+b)
5.\,\frac{\text{d}}{\text{d}x} e^{ax+b} = ae^{ax+b}
6.\,\frac{\text{d}}{\text{d}x} \ln\; (ax+b) = \frac{a}{ax+b}
Product Rule
\dfrac{\text{d}}{\text{d}x}(uv) = u\dfrac{\text{d}v}{\text{d}x} + v\dfrac{\text{d}u}{\text{d}x}
Quotient Rule
\dfrac{\text{d}}{\text{d}x}\left(\dfrac{u}{v}\right)=\dfrac{v\frac{\text{d}u}{\text{d}x}-u\frac{\text{d}v}{\text{d}x}}{v^2}
Gradient Of Curve
From y=\text{f}(x),
\dfrac{\text{d}y}{\text{d}x} is the gradient of the curve.
Gradient Of Tangent & Normal
From \dfrac{\text{d}y}{\text{d}x}, substitute x=k to get m (gradient of tangent).
-\dfrac{1}{m} is the gradient of the normal.
Increasing & Decreasing Functions
1. For increasing functions, \dfrac{\text{d}y}{\text{d}x}>0\$. 2. For decreasing functions,\dfrac{\text{d}y}{\text{d}x}<0.[/stextbox]  [stextbox caption="Rates Of Change" stextbox id="info"]\dfrac{\text{d}y}{\text{d}t}=\dfrac{\text{d}y}{\text{d}x}\times\dfrac{\text{d}x}{\text{d}t}[/stextbox]  [stextbox caption="First Derivative Test" stextbox id="default"]If\dfrac{\text{d}y}{\text{d}x}=0forx=k, test fork^-,k,k^+

    . Maximum point: <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="https://teach.sg/wp-content/ql-cache/quicklatex.com-9f9378ea29ee6f36752bf951c2f47e82_l3.png" height="61" width="142" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & + & 0 & - \\ \hline \end{array}\]" title="Rendered by QuickLaTeX.com"/> Minimum point: <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="https://teach.sg/wp-content/ql-cache/quicklatex.com-e34eb65dd85078db31a0864a230b4e33_l3.png" height="61" width="142" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & - & 0 & + \\ \hline \end{array}\]" title="Rendered by QuickLaTeX.com"/> Inflexion point: <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="https://teach.sg/wp-content/ql-cache/quicklatex.com-6c08639adf6fd54c2707aa0c3de17366_l3.png" height="98" width="142" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & + & 0 & + \\\hline \dfrac{\text{d}y}{\text{d}x} & - & 0 & -  \\\hline \end{array}\]" title="Rendered by QuickLaTeX.com"/>[/stextbox] [stextbox caption="Second Derivative Test" stextbox id="download"]1. If

\dfrac{\text{d}^2y}{\text{d}x^2}<0\$, it is a maximum point.
2. If \dfrac{\text{d}^2y}{\text{d}x^2}>0, it is a minimum point.
3. If \dfrac{\text{d}^2y}{\text{d}x^2}=0, need to do first derivative test.

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