Here you will learn what is derivative or differentiation and various differentiation formulas class 12.
Let’s begin –
What is Derivative or Differentiation ?
Let f(x) be a differentiable or derivable function on [a, b]. Then,
\(lim_{h \to 0}\) \(f(x + h) – f(x)\over h\) or, \(lim_{h \to 0}\) \(f(x – h) – f(x)\over -h\)
is called the derivative or differentiation of f(x) with respect to x and is denoted by
f'(x) or, \(d\over dx\) (f(x)) or, Df(x), where D = \(d\over dx\)
Sometimes the derivative or differentiation of the function f(x) is called the differential coefficient of f(x). The process of finding the derivative of a function by using the above definition is called the differentiation from first principles or by ab-initio method or by delta method.
Differentiation Formulas Class 12
Following are deivatives or differentiation of some standard functions.
(i) \(d\over dx\) \(x^n\) = \(nx^{n-1}\)
(ii) \(d\over dx\) \(e^x\) = \(e^x\)
(iii) \(d\over dx\) \(a^x\) = \(a^xlog_e a\)
(iv) \(d\over dx\) (sin x) = cos x
(v) \(d\over dx\) (cos x) = – sin x
(vi) \(d\over dx\) (tan x) = \(sec^2 x\)
(vii) \(d\over dx\) (cot x) = \(- cosec^2 x\)
(viii) \(d\over dx\) (sec x) = sec x tan x
(ix) \(d\over dx\) (cosec x) = – cosec x cot x
(x) \(d\over dx\) \(sin^{-1} x\) = \(1\over {\sqrt{1 – x^2}}\)
(xi) \(d\over dx\) \(cos^{-1} x\) = – \(1\over {\sqrt{1 – x^2}}\)
(xii) \(d\over dx\) \(tan^{-1} x\) = \(1\over {1 + x^2}\)
(xiii) \(d\over dx\) \(cot^{-1} x\) = -\(1\over {1 + x^2}\)
(xiv) \(d\over dx\) \(sec^{-1} x\) = \(1\over {| x |\sqrt{x^2 – 1}}\)
(xv) \(d\over dx\) \(cosec^{-1} x\) = – \(1\over {| x |\sqrt{x^2 – 1}}\)
(xvi) \(d\over dx\) \(log_e x\) = \(1\over x\)
(xvii) \(d\over dx\) \(log_a x\) = \(1\over xlog_e a\)