List of Derivative Formulas | Differentiation Formulas

In this blog post, we will list all the important formulas of derivatives along with its properties. 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.

Definition of Derivatives in Calculus

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:
$f'(x)=\dfrac{d}{dx}(f(x))=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}.$
Here the prime $’$ denotes the derivative symbol.

List of Differentiation Formulas

The list of differentiation formulas depending upon the function types are provided below.

Basic Differentiation Formulas

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

Trigonometric Functions Derivative Formulas

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

Hypertrigonometric Functions Derivative Formulas

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

Inverse Trigonometric Functions Derivative Formulas

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

Properties of Derivatives

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