Differentiation Formulas | Derivative Formulas List

Math Formulas

A Differentiation formulas list has been provided here for students so that they can refer to these to solve problems based on differential equations. This is one of the most important topics in higher class Mathematics. The general representation of the derivative is d/dx.

Differentiation Formulas List

Some of the general differentiation formulas are;

📘 General Differentiation Formulas

  1. Power Rule: \( \frac{d}{dx}(x^n) = n x^{n-1} \)
  2. Sum Rule: \( \frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x)) \)
  3. Difference Rule: \( \frac{d}{dx}(f(x) – g(x)) = \frac{d}{dx}(f(x)) – \frac{d}{dx}(g(x)) \)
  4. Product Rule: \( \frac{d}{dx}(f(x) \cdot g(x)) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \)
  5. Quotient Rule: \( \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x) \cdot g(x) – f(x) \cdot g'(x)}{g(x)^2} \)
  6. Chain Rule: \( \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) \)
  7. Exponential Function Rule: \( \frac{d}{dx}(e^x) = e^x \)
  8. Exponential Function Rule: \( \frac{d}{dx}(a^x) = a^x \ln a \), where \(a > 0\)
  9. Logarithmic Function Rule (Natural Log): \( \frac{d}{dx}(\ln x) = \frac{1}{x} \)
  10. Logarithmic Function Rule (Base \(a\)): \( \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} \), where \(a > 0\)

Differentiation Formulas

📘 Basic Trigonometric Functions

  1. \( \frac{d}{dx}(\sin x) = \cos x \)
  2. \( \frac{d}{dx}(\cos x) = -\sin x \)
  3. \( \frac{d}{dx}(\tan x) = \sec^2 x \)
  4. \( \frac{d}{dx}(\cot x) = -\text{cosec}^2 x \)
  5. \( \frac{d}{dx}(\sec x) = \sec x \cdot \tan x \)
  6. \( \frac{d}{dx}(\text{cosec} x) = -\text{cosec} x \cdot \cot x \)

📗 Hyperbolic Functions

  1. \( \frac{d}{dx}(\sinh x) = \cosh x \)
  2. \( \frac{d}{dx}(\cosh x) = \sinh x \)
  3. \( \frac{d}{dx}(\tanh x) = \text{sech}^2 x \)
  4. \( \frac{d}{dx}(\coth x) = -\text{cosech}^2 x \)
  5. \( \frac{d}{dx}(\text{sech} x) = -\text{sech} x \cdot \tanh x \)
  6. \( \frac{d}{dx}(\text{cosech} x) = -\text{cosech }x \cdot \text{coth }x \)

📙 Inverse Trigonometric Functions

  1. \( \frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1 – x^2}} \)
  2. \( \frac{d}{dx}(\cos^{-1}x) = \frac{-1}{\sqrt{1 – x^2}} \)
  3. \( \frac{d}{dx}(\tan^{-1}x) = \frac{1}{1 + x^2} \)
  4. \( \frac{d}{dx}(\cot^{-1}x) = \frac{-1}{1 + x^2} \)
  5. \( \frac{d}{dx}(\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2 – 1}} \)
  6. \( \frac{d}{dx}(\text{cosec}^{-1} x) = \frac{-1}{|x|\sqrt{x^2 – 1}} \)

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Differentiation Formulas for Trigonometric Functions

Differentiation Formulas for Hyberbolic Functions

Differentiation Formulas for Inverse Trigonometric Functions

Frequently Asked Questions – FAQs

What is d/dx?

The general representation of the derivative is d/dx. This denotes the differentiation with respect to the variable x.

What is a UV formula?

(d/dx)(uv) = v(du/dx) + u(dv/dx)
This formula is used to find the derivative of the product of two functions.

What are the derivatives of trigonometric functions?

The derivatives of six trigonometric functions are:
(d/dx) sin x = cos x
(d/dx) cos x = -sin x
(d/dx) tan x = sec^2 x
(d/dx) cosec x = -cosec x cot x
(d/dx) sec x = sec x tan x
(d/dx) cot x = -cosec^2 x