Derivative Formulas in Calculus

Derivative Formulas in Calculus – Complete List with Examples

Derivatives are one of the most important concepts in calculus and mathematics. If you are preparing for exams like JEE, IIT, Board Exams, NEET, or university-level math, knowing all derivative formulas is a must. In this post, we will cover basic derivative rules, trigonometric derivatives, exponential and logarithmic derivatives, inverse trigonometric derivatives, and hyperbolic derivatives.

\[\begin{align} & \left( 1 \right)\ \frac{d}{dx}\left( a \right)=0\quad \left[ \text{where }a\text{ is any real constant} \right] \\ & \\ & \left( 2 \right)\ \frac{d}{dx}\left[ f\left( x \right)\pm g\left( x \right)\pm t\left( x \right)\pm .... \right]=f\prime \left( x \right)\pm g\prime \left( x \right)\pm t\prime \left( x \right)\pm .... \\ & \\ & \left( 3 \right)\ \frac{d}{dx}\left[ af\left( x \right) \right]=a\,f\prime \left( x \right)\quad \left[ \text{where }a\text{ is any real constant} \right] \\ & \\ & \left( 4 \right)\ \frac{d}{dx}\left[ u.v \right]=u\frac{dv}{dx}+v\frac{du}{dx}\quad \left[ \text{where }u\text{ and }v\text{ are the functions of }x \right] \\ & \\ & \left( 5 \right)\ \frac{d}{dx}\left[ \frac{u}{v} \right]=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{{{v}^{2}}}\quad \left[ \text{where }v\ne 0\text{ and }u,v\text{ are the functions of }x \right] \\ & \\ & \left( 6 \right)\ \frac{d}{dx}\left( x \right)=1 \\ & \\ & \left( 7 \right)\ \frac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}},\quad n\in \mathbb{R} \\ & \\ & \left( 8 \right)\ \frac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}} \\ & \\ & \left( 9 \right)\ \frac{d}{dx}\left( {{a}^{x}} \right)={{a}^{x}}\,\ln a,\quad \left( a>0,a\ne 1 \right) \\ & \\ & \left( 10 \right)\ \frac{d}{dx}\left( \ln x \right)=\frac{1}{x},\quad \left( x>0 \right) \\ & \\ & \left( 11 \right)\ \frac{d}{dx}\left( {{\log }_{a}}x \right)=\frac{1}{x\ln a},\quad \left( a>0 \right) \\ & \\ & \left( 12 \right)\ \frac{d}{dx}\left( \sin x \right)=\cos x \\ & \\ & \left( 13 \right)\ \frac{d}{dx}\left( \cos x \right)=-\sin x \\ & \\ & \left( 14 \right)\ \frac{d}{dx}\left( \sec x \right)=\sec x\cdot \tan x \\ & \\ & \left( 15 \right)\ \frac{d}{dx}\left( \tan x \right)={{\sec }^{2}}x \\ & \\ & \left( 16 \right)\ \frac{d}{dx}\left( \cos ecx \right)=-\cos ecx\cdot \cot x \\ & \\ & \left( 17 \right)\ \frac{d}{dx}\left( \cot x \right)=\cos e{{c}^{2}}x \\ & \\ & \left( 18 \right)\ \frac{d}{dx}\left( {{\sin }^{-1}}x \right)=\frac{1}{\sqrt{1-{{x}^{2}}}} \\ & \\ & \left( 19 \right)\ \frac{d}{dx}\left( {{\cos }^{-1}}x \right)=-\frac{1}{\sqrt{1-{{x}^{2}}}} \\ & \\ & \left( 20 \right)\ \frac{d}{dx}\left( {{\tan }^{-1}}x \right)=\frac{1}{1+{{x}^{2}}} \\ & \\ & \left( 21 \right)\ \frac{d}{dx}\left( {{\cot }^{-1}}x \right)=-\frac{1}{1+{{x}^{2}}} \\ & \\ & \left( 22 \right)\ \frac{d}{dx}\left( {{\sec }^{-1}}x \right)=\frac{1}{\left| x \right|\sqrt{{{x}^{2}}-1}} \\ & \\ & \left( 23 \right)\ \frac{d}{dx}\left( co{{\sec }^{-1}}x \right)=-\frac{1}{\left| x \right|\sqrt{{{x}^{2}}-1}} \\ &\left( {24} \right)\;\frac{d}{{dx}}\left( {\frac{1}{x}} \right) = - \frac{1}{{{x^2}}}\\ &\left( {25} \right)\;\frac{d}{{dx}}\left( {\frac{1}{{{x^2}}}} \right) = - \frac{2}{{{x^3}}}\\ &\left( {26} \right)\;\frac{d}{{dx}}\left( {\frac{1}{{\sqrt x }}} \right) = \frac{1}{{2\sqrt x }}\\ \end{align}\]  

7. Frequently Asked Questions (FAQ)

Q1: What is the derivative of x²?

Answer: \( \frac{d}{dx}(x^2)=2x \)

Q2: What is the derivative of sin²x?

Answer: Using Chain Rule, \( \frac{d}{dx}(sin^2 x)=2sinx·cosx=sin2x \)

Q3: Why are derivatives important?

Answer: Derivatives are used to find slope, velocity, acceleration, maxima-minima, and optimization in real-life problems.

📌 Conclusion

These are the complete derivative formulas in calculus with rules and examples. Keep this list for quick revision before exams like JEE, NEET, Board Exams, or university tests. If you found this helpful, share it with your friends and bookmark this page.