Derivative Formulas | Differentiation Formulas


List of Derivative Formulas | List of Differentiation Formulas

The concept of the derivative is the backbone of the theory of Calculus. The derivative of a function `f(x)` is defined to be the following limit:
              `f'(x)=frac{d}{dx}(f(x))=lim_{h to 0} frac{f(x+h)-f(x)}{h}.`
Here the prime `’` denotes the derivative symbol. The problems related to differential calculus can be easily solved if you have a complete list of derivative/differential formulas in your table. So we provide here a complete list of basic derivative formulas to help you.

Basic Differentiation Formulas

1. `frac{d}{dx}(x^n)=nx^{n-1} quad` (Power rule of derivatives) 
2. `frac{d}{dx}(c)=0 quad` (`c` is a constant). The derivative of a constant is zero. 
3. `frac{d}{dx}(e^x)=e^x`
4. `frac{d}{dx}(a^x)=a^x log_e a`
5. `frac{d}{dx}(log_e x)=frac{1}{x}` 
6. `frac{d}{dx}(log_a x)=frac{1}{xlog_e a}`

Trigonometric Functions Derivative Formulas

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

Hypertrigonometric Functions Derivative Formulas

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

Inverse Trigonometric Functions Derivative Formulas

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

Properties of Derivatives:

For two differentiable functions `f` and `g` we have: 
Sum/addition Rule of Derivatives: `frac{d}{dx}(f+g)=frac{df}{dx}+frac{dg}{dx}`
Difference/subtraction Rule of Derivatives: `frac{d}{dx}(f-g)=frac{df}{dx}-frac{dg}{dx}`
Product/multiplication Rule of Derivatives: `frac{d}{dx}(fg)=ffrac{dg}{dx}+gfrac{df}{dx}`
Quotient/division Rule of Derivatives: `frac{d}{dx}(frac{f}{g})=frac{g frac{df}{dx}-ffrac{dg}{dx}}{g^2}`
Chain Rule of Derivatives: 
(i) `frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)`
(ii) `frac{dy}{dx}=frac{dy}{du} cdot frac{du}{dx}`

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