Learn rules and formulas for derivative, eliminates the use of limit every time and simplifies the calculation of derivative. Practice derivative power rule, product and quotient rule derivative.
The constant function is an expression which does not involve \(x\) or does not change, if we change the value of \(x\).
Example:
(1) \(\dfrac{d}{dx}(c)=0\) where, \('c'\) is a constant.
The derivative of constant function is 0.
(2) \(\dfrac{d}{dx}(\pi)=0\)
(3) \(\dfrac{d}{dt}(3)=0\)
(4) \(\dfrac{d}{dt}(2)=0\)
(1) \(\dfrac{d}{dx}(x)=1\)
(2) \(\dfrac{d}{dx}(x^n)=nx^{n-1}\) where \(n\) is any positive integer.
\(\dfrac{d}{dx}(x)=\dfrac{d}{dx}\) \(x^1\) = \(1x^0\)
\(=1×1=1\)
Example : \(\dfrac{d}{dx}(x^{20})=20x^{19}\)
(3) \(\dfrac{d}{dx}(x^{15})=15x^{14}\)
(4) \(\dfrac{d}{dx}(x^{100})=100x^{99}\)
(5) \(\dfrac{d}{dx}(x^{250})=250x^{249}\)
\(\dfrac{d}{dx}\left(f(x)-g(x)\right)=\dfrac{d}{dx}f(x)-\dfrac{d}{dx}\left(g(x)\right)\)
(The derivative of difference of two function is the difference of derivative).
\((1)\;\dfrac{d}{dx}(x^5-x^4+x^3-x^2+1)\)
\(=\dfrac{d}{dx}(x^5)-\dfrac{d}{dx}(x^4)+\dfrac{d}{dx}(x^3)-\dfrac{d}{dx}(x^2)+\dfrac{d}{dx}(1)\)
\(=5x^4-4x^3+3x^2-2x^1+0\)
\(=5x^4-4x^3+3x^2-2x\)
\((2)\;\dfrac{d}{dx}(12x^3-4x^2+x-5x^5)\)
\(=\dfrac{d}{dx}(12x^3)-\dfrac{d}{dx}(4x^2)+\dfrac{d}{dx}(x)-\dfrac{d}{dx}(5x^5)\)
\(=12×3x^2-4×2x+1-5×5x^4\)
\(=36x^2-8x+1-25x^4\)
\(=-25x^4+36x^2-8x+1\)
A \(100x^3-27x^2+1\)
B \(25x^7-8x^3+12\)
C \(54x^2-18x\)
D \(27x^{10}+8x^2+7\)
Product Rule
\(\dfrac{d}{dx}\left(f(x)\,g(x)\right)=f(x)\,\dfrac{d}{dx}g(x)+g(x)\dfrac{d}{dx}f(x)\)
\((1)\;\dfrac{d}{dx}(x+2)(x^2-3x)\)
The given function is a product of two functions, \((x+2)\) and \((x^2-3x)\) .
We can take any function as \(f(x)\) and \(g(x)\)
Let \((x+2)=f(x)\;\text{and}\;(x^2-3x)=g(x)\)
Now, applying product rule:
\(\dfrac{d}{dx}(x+2)(x^2-3x)\)
\(=(x+2)\dfrac{d}{dx}(x^2-3x)+(x^2-3x)\dfrac{d}{dx}(x+2)\)
\(=(x+2)\left[\dfrac{d}{dx}(x^2)-\dfrac{d}{dx}(3x)\right]+(x^2-3x)\left[\dfrac{d}{dx}(x)+\dfrac{d}{dx}(2)\right]\)
\(=(x+2)[2x-3x^0]+(x^2-3x)[x^0+0]\)
\(=(x+2)[2x-3]+(x^2-3x)[1]\)
\(=(x+2)[2x-3]+x^2-3x\)
\(=x(2x-3)+2(2x-3)+x^2-3x\)
\(=2x^2-3x+4x-6+x^2-3x\)
\(=3x^2-2x-6\)
A \(14x^2-28x+55\)
B \(19x^3+45x^2+6\)
C \(2x^3+8x^2+18\)
D \(30x^2+38x+22\)
(1) \(\dfrac{d}{dx}\left(cf(x)\right)=c\dfrac{d}{dx}\left(f(x)\right)\) where \('c'\) is a constant
(The derivative of constant times a function is constant times the derivative of function).
\((1)\;\dfrac{d}{dx}(2x^{4})=2\dfrac{d}{dx}(x^4)\) \(\left[\dfrac{d}{dx}x^n=nx^{n-1}\right]\)
\(=2×4×x^{4-1}=8x^3\)
\((2)\;\dfrac{d}{dx}(4x)=4×\dfrac{d}{dx}(x)\)
\(=4×1×x^{1-1}=4x^0=4\)
\((3)\;\dfrac{d}{dx}(6x^{100})=6×\dfrac{d}{dx}(x^{100})\)
\(=6×100\;x^{100-1}\)
\(=600x^{99}\)
\(\dfrac{d}{dx}\,\left(f(x)+g(x)\right)=\dfrac{d}{dx}\,\left(f(x)\right)+\dfrac{d}{dx}\,\left(g(x)\right)\)
(The derivative of a sum of function is the sum of derivatives)
\(\Rightarrow\) \((f+g+h)\)\('\)= \(f'+g'+h'\)
\((1)\;\dfrac{d}{dx}(x^2+2x+1)\)
\(=\dfrac{d}{dx}(x^2)+\dfrac{d}{dx}(2x)+\dfrac{d}{dx}(1)\)
\(=2x^1+2×1x^0+0\)
\(=2x+2\)
\((2)\;\dfrac{d}{dx}\left(3x^4+14x^3+3x^2+2\right)\)
\(=\dfrac{d}{dx}(3x^4)+\dfrac{d}{dx}(14x^3)+\dfrac{d}{dx}(3x^2)+\dfrac{d}{dx}(2)\)
\(=3×4x^3+14×3x^2+3×2x^1+0\)
\(=12x^3+42x^2+6x\)
A \(42x^2+5x^7\)
B \(8x^3+56x^7\)
C \(15x^2-x\)
D \(47x^5-x^8\)
Ratio Rule or the Quotient Rule
\(\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)\dfrac{d}{dx}f(x)-f(x)\dfrac{d}{dx}g(x)}{(g(x))^2}\)
\(\dfrac{d}{dx}\left(\dfrac{\text{High}}{\text{Low}}\right)\)
\(=\dfrac{\text{Low}\frac{d}{dx}\text{(High)}-\text{(High)}\frac{d}{dx}(\text{Low})}{(\text{Low})^2}\)
"Low D high minus high D low cross the line and square the low"
\(\dfrac{d}{dx}\left(\dfrac{x^2+2}{x+1}\right)\)
\(=\dfrac{(x+1)\frac{d}{dx}(x^2+2)-(x^2+2)\frac{d}{dx}(x+1)}{(x+1)^2}\)
\(=\dfrac{(x+1)\left[\frac{d}{dx}(x^2)+\frac{d}{dx}(2)\right]-(x^2+2)\left[\frac{d}{dx}(x)+\frac{d}{dx}(1)\right]}{(x+1)^2}\)
\(=\dfrac{(x+1)[2x+0]-(x^2+2)[1+0]}{(x+1)^2}\)
\(=\dfrac{(x+1)(2x)-(x^2+2)}{(x+1)^2}\)
\(=\dfrac{2x^2+2x-x^2-2}{(x+1)^2}\)
\(=\dfrac{x^2+2x-2}{(x+1)^2}\)
A \(\dfrac{x^4+2x^3+5x^2-2}{(x^2+x+1)^2}\)
B \(\dfrac{x^4+8x^3+x+3}{(x^2+5x+6)^2}\)
C \(\dfrac{2x^2-8x+7}{(5x+1)^2}\)
D \(\dfrac{8x^2+7x+3}{(x^2+x+1)^2}\)
\(\dfrac{d}{dx}(x^n)=nx^{n-1}\) for all values of \(n\) .
Example:
\((1)\;\dfrac{d}{dx}\left(\dfrac{1}{\sqrt x^3}\right)\)
\(=\dfrac{d}{dx}\left(x^\frac{-3}{2}\right)\)
\(=\dfrac{-3}{2}x^{-3/2-1}\)
\(=\dfrac{-3}{2}x^{-5/2}\)
\((2)\;\dfrac{d}{dx}\left(\dfrac{2\sqrt x}{x^{\frac{4}{5}}}\right)\)
\(=\dfrac{d}{dx}\left(2\dfrac{x^{\frac{1}{2}}}{x^{\frac{4}{5}}}\right)\)
\(=\dfrac{d}{dx}\left(2x^{\frac{1}{2}-\frac{4}{5}}\right)\)
\(=\dfrac{d}{dx}\left(2x^{\frac{5-8}{10}}\right)\)
\(=\dfrac{d}{dx}\left(2x^{\frac{-3}{10}}\right)\)
\(=2×\left(\dfrac{-3}{10}\right)x^{\frac{-3}{10}-1}\)
\(=\dfrac{-3}{5}x^{\frac{-3-10}{10}}\)
\(=\dfrac{-3}{5}x^{\frac{-13}{10}}\)
A \(\dfrac{2}{x^2(\sqrt x+1)}\)
B \(\dfrac{1}{\sqrt x(\sqrt x+1)^2}\)
C \(\dfrac{5x}{( x+1)^2}\)
D \(\dfrac{3x^3}{x^2+1}\)