Find the derivative of \(x^{2}-2\) at \(x=10\).
20
Find the derivative of \(x\) at \(x=1\).
1
Find the derivative of \(99x\) at \(x=100\).
99
Find the derivative of the following functions from first principle:
(i) \(x^{3} - 27\)
(ii) \((x-1)(x-2)\)
(iii) \(\dfrac{1}{x^{2}}\)
(iv) \(\dfrac{x+1}{x-1}\)
(i) \(3x^{2}\)
(ii) \(2x - 3\)
(iii) \(-\dfrac{2}{x^{3}}\)
(iv) \(-\dfrac{2}{(x-1)^{2}}\)
For the function
\(f(x)=\dfrac{x^{100}}{100}+\dfrac{x^{99}}{99}+\cdots+\dfrac{x^{2}}{2}+x+1\),
prove that \(f'(1)=100f'(0)\).
Result: \(f'(1)=100f'(0)\)
Find the derivative of the expression:
\(x^{n}+ax^{n-1}+a^{2}x^{n-2}+\cdots +a^{n-1}x+a^{n}\)
\(nx^{n-1}+a(n-1)x^{n-2}+a^{2}(n-2)x^{n-3}+\cdots +a^{n-1}\)
For some constants \(a\) and \(b\), find the derivative of the following:
(i) \((x-a)(x-b)\)
(ii) \((ax^{2}+b)^{2}\)
(iii) \(\dfrac{x-a}{x-b}\)
(i) \(2x - a - b\)
(ii) \(4ax(ax^{2}+b)\)
(iii) \(\dfrac{a-b}{(x-b)^{2}}\)
Find the derivative of
\(\dfrac{x^{n}-a^{n}}{x-a}\)
for constant \(a\).
\(\dfrac{nx^{n}-anx^{n-1}-x^{n}+a^{n}}{(x-a)^{2}}\)
Find the derivative of the following:
(i) \(2x - \dfrac{3}{4}\)
(ii) \((5x^{3}+3x-1)(x-1)\)
(iii) \(x^{-3}(5+3x)\)
(iv) \(x^{5}(3-6x^{-9})\)
(v) \(x^{-4}(3-4x^{-5})\)
(vi) \(\dfrac{2}{x+1} - \dfrac{x^{2}}{3x-1}\)
(i) 2
(ii) \(20x^{3} - 15x^{2} + 6x - 4\)
(iii) \(-\dfrac{3}{x^{4}}(5+2x)\)
(iv) \(15x^{4} + \dfrac{24}{x^{5}}\)
(v) \(-\dfrac{12}{x^{5}} + \dfrac{36}{x^{10}}\)
(vi) \(-\dfrac{2}{(x+1)^{2}} - \dfrac{x(3x-2)}{(3x-1)^{2}}\)
Find the derivative of \(\cos x\) from first principle.
-\sin x
Find the derivative of the following functions:
(i) \(\sin x \cos x\)
(ii) \(\sec x\)
(iii) \(5\sec x + 4\cos x\)
(iv) \(\csc x\)
(v) \(3\cot x + 5\cosec x\)
(vi) \(5\sin x - 6\cos x + 7\)
(vii) \(2\tan x - 7\sec x\)
(i) \(\cos 2x\)
(ii) \(\sec x \tan x\)
(iii) \(5\sec x \tan x - 4\sin x\)
(iv) \(-\csc x \cot x\)
(v) \(-3\csc^{2}x - 5\csc x \cot x\)
(vi) \(5\cos x + 6\sin x\)
(vii) \(2\sec^{2}x - 7\sec x \tan x\)