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} \text{ 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} \text{ is the gradient of the normal.}

🦉 Increasing & Decreasing Functions

\text{1. For increasing functions, } \dfrac{\text{d}y}{\text{d}x}>0

\text{2. For decreasing functions, } \dfrac{\text{d}y}{\text{d}x}<0

🦉 Rates Of Change

\dfrac{\text{d}y}{\text{d}t}=\dfrac{\text{d}y}{\text{d}x}\times\dfrac{\text{d}x}{\text{d}t}

🦉 First Derivative Test

If $\dfrac{\text{d}y}{\text{d}x}=0$ for $x=k$, test for $k^-$, $k$, $k^+$.

Maximum point:

\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & + & 0 & - \\ \hline \end{array}

Minimum point:

\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & - & 0 & + \\ \hline \end{array}

Inflexion point:

\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}

🦉 Second Derivative Test

\text{1. If } \dfrac{\text{d}^2y}{\text{d}x^2}<0, \text{ it is a maximum point.}

\text{2. If } \dfrac{\text{d}^2y}{\text{d}x^2}>0, \text{ it is a minimum point.}

\text{3. If } \dfrac{\text{d}^2y}{\text{d}x^2}=0, \text{ need to do first derivative test.}