Differentiation Formulas - Quotient Rule, Chain Rule, List of Formulas and Inverse Trigonometric Functions

Differentiation

In mathematics, differentiation is a part of calculus that is defined as the derivative of any function using an independent variable. It can also be referred to as the function per unit change. It is denoted by dy/dx or y’, where x is the independent variable and y is differentiated concerning x. If any function has an extremely small change ‘k’ close to x, then the derivative is known as lim k→0 {f(x+k)–f(x) / k} There are rudimentary four types of differentiation rules: Product Rule: It states that the differentiation of the product of two functions is the sum of the product of the second function with the derivative of the first function and the first function with the derivative of the second function. d/dx (pq) = q * dp/dx + p * dq/dx Sum and Difference Rule: This rule states that if two functions are added or subtracted, then their differentiation is also added or subtracted. F(X) () F(Y) = F’(X) () F’(Y)

Quotient Rule: It states that the differentiation of the ratio of two functions is the difference of product of denominator function with the derivative of numerator function and numerator function with the derivative of denominator function divided by the square of the denominator function. d/dx {p(x) / q(x)} = [q(x) * p’(x) – p(x) * q’(x)]/ p(x)2

Chain Rule: This rule is used to determine the differentiation of composite function. The concept is to solve the outer function first and eventually solve the innermost function. For instance, cos(a²) is a composite function because it can be constructed as p(q(x)) for p(x) = cos(a) and q(x)=a². For solving a lot of differentiations easily, differentiation formulas are derived and provided. To ease the solving processes, differentiation of trigonometric functions, logarithmic functions, exponential functions, hyperbolic functions, etc. are predefined. The list of formulas is given below:

d/dp (sin p)	cos p
d/dp (cos p)	–sin p
d/dp (tan p)	sec²p
d/dp (cot p)	−cosec²p
d/dp (sec p)	sec p tan p
d/dp (cosec p)	−cosec p cot p
d/dp (sinh p)	cosh p
d/dp (cosh p)	sinh p
d/dp (tanh p)	sech²p
d/dp (coth p)	−cosech²p
d/dp (sech p)	−sech p tanh p
d/dp (cosech p)	−cosech p coth p

Inverse Trigonometric Functions
d/dp (sin−1 p)	1 /√1–p²
d/dp (cos−1 p)	−1/ √1–p²
d/dp (tan−1 p)	1/1+p²
d/dp (cot−1 p)	−1/1+p²
d/dp (sec−1 p)	1/√\|p\|p²–1
d/dp (cosec−1 p)	−1/√\|p\|p²–1

Other Differentiation Formulas
d/dp (loga p)	1/ (ln a) p
d/dp (ln p)	1/p
d/dp(e^p)	e^p
d/dp(a^p)	ap ln(a)

These formulas can be directly substituted to solve many equations without proving these first. They are standard results derived by great mathematicians. You can use the first principle of differentiation to prove these formulas if you wish to. These must be memorized because they are crucial in answering the bulk of questions. Only basic functions are covered by these rules. These derivatives will not work if the function's equation grows complex. The chain rule discussed earlier is used to solve complicated derivatives.