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 derivatives or differentiation of some standard functions.
Basic Differentiation Formulas
(i) \(d\over dx\) \(x^n\) = \(nx^{n-1}\)
(ii) \(d\over dx\)(a) = 0, where a is constant.
(iii) \(d\over dx\)(x) = 1
(iv) \(d\over dx\)(kx) = k, where k is constant
Differentiation of Logarithmic and Exponential Function Formulas
(i) \(d\over dx\) \(e^x\) = \(e^x\)
(ii) \(d\over dx\) \(a^x\) = \(a^xlog_e a\)
(iii) \(d\over dx\) \(log_e x\) = \(1\over x\)
(iv) \(d\over dx\) \(log_a x\) = \(1\over xlog_e a\)
Trigonometric Function Differentiation Formulas Class 12
(i) \(d\over dx\) (sin x) = cos x
(ii) \(d\over dx\) (cos x) = – sin x
(iii) \(d\over dx\) (tan x) = \(sec^2 x\)
(iv) \(d\over dx\) (cot x) = \(- cosec^2 x\)
(vi) \(d\over dx\) (sec x) = sec x tan x
(vi) \(d\over dx\) (cosec x) = – cosec x cot x
Inverse Trigonometric Function Differentiation Formulas
(i) \(d\over dx\) \(sin^{-1} x\) = \(1\over {\sqrt{1 – x^2}}\)
(ii) \(d\over dx\) \(cos^{-1} x\) = – \(1\over {\sqrt{1 – x^2}}\)
(iii) \(d\over dx\) \(tan^{-1} x\) = \(1\over {1 + x^2}\)
(iv) \(d\over dx\) \(cot^{-1} x\) = -\(1\over {1 + x^2}\)
(v) \(d\over dx\) \(sec^{-1} x\) = \(1\over {| x |\sqrt{x^2 – 1}}\)
(vi) \(d\over dx\) \(cosec^{-1} x\) = – \(1\over {| x |\sqrt{x^2 – 1}}\)
Differentiation Rules Class 12
(i) Product Rule – \(d\over dx\) {f(x) g(x)} = \(d\over dx\) (f(x)) g(x) + f(x). \(d\over dx\) (g(x))
(ii) Quotient Rule – \({g(x) {d\over dx} (f(x)) – f(x) {d\over dx} (g(x))}\over {(g(x))^2}\)
(iii) Chain Rule – \(d\over dx\) {(fog) (x)} = \(d\over d g(x)\) {(fog) (x)} \(d\over dx\) (g(x)).
