Essential Differentiation Formulas for Calculus Students

Differentiation Formulas

Differentiation, a fundamental concept in calculus, involves computing the rate at which things change. To aid students, especially in higher classes such as 11th and 12th, a comprehensive list of differentiation formulas is essential. This helps in solving a variety of problems from the syllabus effectively. Here, we’ll explore these formulas, focusing on their application to different functions.

General Differentiation Formulas

Differentiation can transform complex dynamic situations into more manageable ones by breaking them down into simpler parts. Here’s how some basic operations are handled in differentiation:

Power Rule: The derivative of $\boldsymbol{x^n}$ with respect to x is $\boldsymbol{nx^{n-1}}$
Constant Rule: The derivative of a constant is 0.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant times the derivative of the function.
Sum Rule: The derivative of a sum or difference is the sum or difference of the derivatives.
Product Rule: The derivative of a product of two functions is $\boldsymbol{f'g + fg'}$
Quotient Rule: The derivative of a quotient of two functions is $\boldsymbol{\frac{f'g - fg'}{g^2}}$

Differentiation of Trigonometric Functions

Trigonometry deals with the relationships between the angles and sides of triangles. The derivative rules for the basic trigonometric functions are as follows:

$\boldsymbol{1. \quad \frac{d}{dx} (sin x) = cos x}$

$\boldsymbol{2. \quad \frac{d}{dx} (cos x) = -sin x}$

$\boldsymbol{3. \quad \frac{d}{dx} (tan x) = sec^2 x}$

$\boldsymbol{4. \quad \frac{d}{dx} (cot x) = -csc^2 x}$

$\boldsymbol{5. \quad \frac{d}{dx} (sec x) = sec x tan x}$

$\boldsymbol{6. \quad \frac{d}{dx} (csc x) = -csc x cot x}$

$\boldsymbol{7. \quad \frac{d}{dx} (sinh x) = cosh x}$

$\boldsymbol{8. \quad \frac{d}{dx} (cosh x) = sinh x}$

$\boldsymbol{9. \quad \frac{d}{dx} (tanh x) = sech^2 x}$

$\boldsymbol{10. \quad \frac{d}{dx} (coth x) = -csch^2 x}$

$\boldsymbol{11. \quad \frac{d}{dx} (sech x) = -sech x tanh x}$

$\boldsymbol{12. \quad \frac{d}{dx} (csch x) = -csch x coth x}$

$\boldsymbol{sin(x)}$ : The derivative is $\boldsymbol{cos(x)}$
$\boldsymbol{cos(x)}$ : The derivative is − $\boldsymbol{sin(x)}$
$\boldsymbol{tan(x)}$ : The derivative is $\boldsymbol{sec^2(x)}$
The derivatives for other functions like cotangent, secant, and cosecant follow similar trigonometric identities.

Differentiation of Hyperbolic Functions

Hyperbolic functions, which are analogs of trigonometric functions but for a hyperbola, also have specific differentiation rules:

$\boldsymbol{sinh(x)}$ : Derivative is $\boldsymbol{cosh(x)}$
$\boldsymbol{cosh(x)}$ : Derivative is $\boldsymbol{sinh(x)}$

Differentiation of Inverse Trigonometric Functions

Inverse trigonometric functions allow angles to be derived from ratio values, and their differentiation is pivotal in calculus:

$\boldsymbol{1. \quad \frac{d}{dx} (sin^{-1} x) = \frac{1}{sqrt{1 - x^2}}}$

$\boldsymbol{2. \quad \frac{d}{dx} (cos^{-1} x) = \frac{-1}{sqrt{1 - x^2}}}$

$\boldsymbol{3. \quad \frac{d}{dx} (tan^{-1} x) = \frac{1}{1 + x^2}}$

$\boldsymbol{4. \quad \frac{d}{dx} (cot^{-1} x) = \frac{-1}{1 + x^2}}$

$\boldsymbol{5. \quad \frac{d}{dx} (sec^{-1} x) = \frac{1}{|x|sqrt{x^2 - 1}}}$

$\boldsymbol{6. \quad \frac{d}{dx} (csc^{-1} x) = \frac{-1}{|x|sqrt{x^2 - 1}}}$

$\boldsymbol{sin^{-1}(x)}$ : Derivative is $\boldsymbol{\frac{1}{sqrt{1 - x^2}}}$
$\boldsymbol{cos^{-1}(x)}$ : Derivative is $\boldsymbol{\frac{-1}{sqrt{1 - x^2}}}$
$\boldsymbol{tan^{-1}(x)}$ : Derivative is $\boldsymbol{\frac{1}{1 + x^2}}$

Additional Differentiation Rules

$\boldsymbol{1. \quad \frac{d}{dx} (a^x) = a^x ln a}$

$\boldsymbol{2. \quad \frac{d}{dx} (e^x) = e^x}$

$\boldsymbol{3. \quad \frac{d}{dx} (log_a x) = \frac{1}{(ln a)x}}$

$\boldsymbol{4. \quad \frac{d}{dx} (ln x) = \frac{1}{x}}$

$\boldsymbol{5. \text{Chain Rule:}}$

$\boldsymbol{\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = \frac{dy}{dv} \times \frac{dv}{du} \times \frac{du}{dx}}$

Chain Rule: Used to differentiate composite functions, the derivative of $\boldsymbol{f(g(x))}$ is $\boldsymbol{f'(g(x))g'(x)}$ .

This overview not only helps in academic applications but also in understanding the changes and rates at which they occur in various scientific and engineering fields.