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Here is the list of Class 12 differentiation formulas to remember to score well in your Class 12 Mathematics Board examination. The formula list include the continuity condition ,Differentiability condition, inverse trigonometric function, Logarithm function, exponential function. This also includes the rules for finding the derivative of various composite function using chain rule, Logarithmic Differentiation.
Continuity
- A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point.
- A function is continuous if it is continuous on the whole of its domain
- Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions
Differentiable at a point a
$\lim_{h \rightarrow 0 -0} \frac{f(a+h) -f(a)}{h} = \lim_{h \rightarrow 0 +0} \frac{f(a+h) -f(a)}{h}= \text {finite number}$
Important Points on Differentiability
- A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b]. At at the end points a and b, we take the right hand limit and left hand limit, which are nothing but left hand derivative and right hand derivative of the function at a and b respectively
- Every differentiable function is continuous, but the converse is not true.
Standard Differentiation formulas
$\frac {d}{dx} (e^x) = e^x$
$\frac {d}{dx} (ln x) = \frac {1}{x}$
$\frac {d}{dx} (log_{10} x) =\frac {1}{x ln 10}$
$\frac {d}{dx} (log_{a} x) =\frac {1}{x ln a}$
$\frac {d}{dx} (a^x) = a^x ln a$
Differentiation formulas for Inverse Trigonometric Functions
$\frac {d}{dx} (sin^{-1}x) = \frac {1}{\sqrt {1-x^2} }, -1< x< 1$
$\frac {d}{dx} (cos ^{-1}x) = -\frac {1}{\sqrt {1-x^2}} , -1< x< 1$
$\frac {d}{dx} (tan ^{-1}x) = \frac {1}{1 + x^2}$
$\frac {d}{dx} (cot ^{-1}x) = -\frac {1}{1 + x^2}$
$\frac {d}{dx} (sec ^{-1}x) = \frac {1}{|x|\sqrt {x^-1}} , |x| > 1$
$\frac {d}{dx} (cosec ^{-1}x) = \frac {1}{|x|\sqrt {x^-1}} , |x| > 1$$
Chain Rule
if y = f(u) and u =g(x) ,then
$\frac {dy}{dx} = \frac {dy}{du} \frac {du}{dx}$
Example
$\frac {d}{dx} [sin (x^3)] = \frac {d}{du} sin (u) \frac {d}{dx} (x^3)= 3x^2 cos (x^3)$
Logarithmic Differentiation
If $ y = \frac {f_1(x) f_2(x) f_3(x)}{ g_1(x) g_2(x) g_3(x)}$
Then we first take logarithm and then differentiate it
nth Derivatives of Common Function
$\frac {d^n}{dx^n} [sin(ax + b)]= a^n sin (\frac {n\pi}{2} + ax + b)$
$\frac {d^n}{dx^n} [cos(ax + b)]= a^n cos (\frac {n\pi}{2} + ax + b)$
$\frac {d^n}{dx^n} [e^{ax}]= a^n e^{ax}$
Parametric Form
let y=f(t), x= g(t)
$\frac {dy}{dx} = \frac {\frac {dy}{dt}}{ \frac {dx}{dt}}$
Second Order Derivative
If y = f(x)
then
$\frac {dy}{dx} = f'(x)$
Second Order Derivative is obtain by again differentiating with respect to x
$\frac {d}{dx} \frac {dy}{dx} = \frac {d}{dx} f'(x)$
$\frac {d^2 y}{dx^2} = f”x $
