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:
| $\dfrac{d}{dx}(f+g)=\dfrac{df}{dx}+\dfrac{dg}{dx}$ |
Difference/subtraction Rule of Derivatives:
| $\dfrac{d}{dx}(f-g)=\dfrac{df}{dx}-\dfrac{dg}{dx}$ |
Product/multiplication Rule of Derivatives:
| $\dfrac{d}{dx}(fg)=f\dfrac{dg}{dx}+g\dfrac{df}{dx}$ |
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) $\dfrac{dy}{dx}=\dfrac{dy}{du} \cdot \dfrac{du}{dx}$
|