Differential Calculus ~ My Profile

Power Rule: When we need to find the derivative of an exponential function, the power rule states that:

\frac{d}{d x} x^{n} = n \times x^{n - 1}

Product Rule: When

f (x)

is the product of two functions,

a (x)

and

b (x)

, then the product rule states that:

\frac{d}{d x} f (x) = \frac{d}{d x} [a (x) \times b (x)] = b (x) \times \frac{d}{d x} a (x) + a (x) \times \frac{d}{d x} b (x)

Quotient Rule: When

f (x)

is of the form

\frac{a (x)}{b (x)}

, then the quotient rule states that:

\frac{d}{d x} f (x) = \frac{d}{d x} [\frac{a (x)}{b (x)}] = \frac{b (x) \times \frac{d}{d x} a (x) - a (x) \times \frac{d}{d x} b (x)}{{[b (x)]}^{2}}

Sum or Difference Rule: When a function

f (x)

is the sum or difference of two functions

a (x)

and

b (x)

, then the sum or difference formula states that:

\frac{d}{d x} f (x) = \frac{d}{d x} [a (x) \pm b (x)] = \frac{d}{d x} a (x) \pm \frac{d}{d x} b (x)

Derivative of a Constant: Derivative of a constant is always zero.
Suppose

f (x) = c

, where c is a constant. We have,

\frac{d}{d x} f (x) = \frac{d}{d x} (c) = 0

Derivative of a Constant Multiplied with a Function

f

: When we need to find out the derivative of a constant multiplied with a function, we apply this rule:

\frac{d}{d x} [c \times f (x)] = c \times \frac{d}{d x} f (x)

Chain Rule: The chain rule of differentiation states that:

dydx=dydu×dudx

Differentiation Formula List

(i)

\frac{d}{d x} (k) = 0

(ii)

\frac{d}{d x} (k u) = k \frac{d u}{d x}

(iii)

\frac{d}{d x} (u \pm v) = \frac{d u}{d x} \pm \frac{d v}{d x}

(iv)

\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}

(v)

\frac{d}{d x} (u / v) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}

(vi)

\frac{d y}{d x} . \frac{d x}{d y} = 1

(vii)

\frac{d}{d x} (x^{n}) = n x^{n - 1}

(viii)

\frac{d}{d x} (e^{x}) = e^{x}

(ix)

\frac{d}{d x} (a^{x}) = a^{x} \log a

(x)

\frac{d}{d x} (\log x) = \frac{1}{x}

(xi)

\frac{d}{d x} \log_{a} x = \frac{1}{x} \log_{a} e

(xii)

\frac{d^{n}}{d x^{n}} (a x + b)^{n} = n! a^{n}

Differentiation for Trignometric Functions

(i)

\frac{d}{d x} (\sin x) = \cos x

(ii)

\frac{d}{d x} (\cos x) = - \sin x

(iii)

\frac{d}{d x} (\tan x) = \sec^{2} x

(iv)

\frac{d}{d x} (\cot x) = - c o s e c^{2} x

(v)

\frac{d}{d x} (\sec x) = \sec x \tan x

(vi)

\frac{d}{d x} (c o s e c x) = - c o s e c x \cot x

(vii)

\frac{d}{d x} (\sin u) = \cos u \frac{d u}{d x}

(viii)

\frac{d}{d x} (\cos u) = - \sin u \frac{d u}{d x}

(ix)

\frac{d}{d x} (\tan u) = \sec^{2} u \frac{d u}{d x}

(x)

\frac{d}{d x} (\cot u) = - c o s e c^{2} u \frac{d u}{d x}

(xi)

\frac{d}{d x} (\sec u) = \sec u \tan u \frac{d u}{d x}

(xii)

\frac{d}{d x} (c o s e c u) = - c o s e c u \cot u \frac{d u}{d x}

Differentiation Formulas For Inverse Trigonometric Functions

(i)

\frac{d}{d x} (\sin^{- 1} x)

\frac{1}{\sqrt{1 - x^{2}}}

(ii)

\frac{d}{d x} (\cos^{- 1} x)

= -

\frac{1}{\sqrt{1 - x^{2}}}

(iii)

\frac{d}{d x} (\tan^{- 1} x)

\frac{1}{1 + x^{2}}

(iv)

\frac{d}{d x} (\cot^{- 1} x)

= -

\frac{1}{1 + x^{2}}

(v)

\frac{d}{d x} (\sec^{- 1} x)

\frac{1}{x \sqrt{x^{2} - 1}}

(vi)

\frac{d}{d x} (c o s e s^{- 1} x)

= -

\frac{1}{x \sqrt{x^{2} - 1}}

(vii)

\frac{d}{d x} (\sin^{- 1} u)

\frac{1}{\sqrt{1 - u^{2}}} \frac{d u}{d x}

(viii)

\frac{d}{d x} (\cos^{- 1} u)

= -

\frac{1}{\sqrt{1 - u^{2}}} \frac{d u}{d x}

(ix)

\frac{d}{d x} (\tan^{- 1} u)

\frac{1}{1 + u^{2}} \frac{d u}{d x}

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

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Monday, 3 May 2021