105 LearnersLast updated on August 6th, 2025
Math Formula for Derivatives
In calculus, derivatives represent how a function changes as its input changes. They are fundamental in understanding rates of change and slopes of curves. In this topic, we will learn the formulas for finding derivatives.
List of Math Formulas for Derivatives
Derivatives are used to calculate the rate of change of a function with respect to a variable. Let’s learn the basic formulas for calculating derivatives.
Derivative Formula for Basic Functions
The derivative of a constant function is 0.
For a linear function f(x)=ax+b, the derivative is f'(x)=a.
For a power function f(x)=x^n, the derivative is f'(x)=nx^(n-1).
Derivative Formula for Trigonometric Functions
The derivative of sin(x) is cos(x).
The derivative of cos(x) is -sin(x).
The derivative of tan(x) is sec^2(x).
Derivative Formula for Exponential and Logarithmic Functions
The derivative of e^x is e^x.
The derivative of a^x (where a is a constant) is a^x ln(a).
The derivative of ln(x) is 1/x.
Importance of Derivative Formulas
In math and real life, derivative formulas are crucial for analyzing and understanding changes. Here are some important uses of derivatives:
- Derivatives help in determining the slope or rate of change.
- They are instrumental in finding maxima and minima of functions.
- In physics, derivatives are used to calculate velocity and acceleration.
Tips and Tricks to Memorize Derivative Formulas
Students often find derivative formulas tricky. Here are some tips to master them:
- Use simple mnemonics like "derivative of sin is cos" for trigonometric functions.
- Connect derivative formulas with real-life scenarios, such as speed and growth rates.
- Practice regularly by solving problems and revising the formulas.
Common Mistakes and How to Avoid Them While Using Derivative Formulas
Students make errors when calculating derivatives. Here are some mistakes and how to avoid them:
Mistake 1
Not applying the power rule correctly
Students sometimes apply the power rule incorrectly, especially when the exponent is negative or fractional. Always remember to decrease the exponent by one and multiply by the original exponent.
Mistake 2
Forgetting chain rule in composite functions
When dealing with composite functions, students often forget to apply the chain rule. Always differentiate the outer function and multiply by the derivative of the inner function.
Mistake 3
Ignoring negative signs in trigonometric derivatives
Students may ignore negative signs when differentiating functions like cos(x) or cot(x). Always pay attention to signs to avoid errors.
Mistake 4
Confusing derivative rules for different functions
Students often mix up rules for different types of functions. Practice regularly to keep the rules clear in your mind.
Mistake 5
Skipping the product or quotient rule
Students may forget to use the product or quotient rule when differentiating products or quotients of functions. Always identify these situations and apply the appropriate rule.
Examples of Problems Using Derivative Formulas
Problem 1
Find the derivative of f(x)=3x^4.
The derivative is f'(x)=12x^3.
Explanation
Using the power rule:
f(x)=3x^4 f'(x)=3*4x^(4-1)=12x^3.
Problem 2
Find the derivative of f(x)=sin(x).
The derivative is f'(x)=cos(x).
Explanation
The derivative of sin(x) with respect to x is cos(x).
Problem 3
Find the derivative of f(x)=e^x.
The derivative is f'(x)=e^x.
Explanation
The derivative of e^x with respect to x is e^x.
Problem 4
Find the derivative of f(x)=ln(x).
The derivative is f'(x)=1/x.
Explanation
The derivative of ln(x) with respect to x is 1/x.
Problem 5
Find the derivative of f(x)=5x.
The derivative is f'(x)=5.
Explanation
For the linear function f(x)=5x, the derivative is simply the coefficient of x, which is 5.
FAQs on Derivative Formulas
1.What is the derivative formula for a power function?
2.What is the formula for the derivative of sin(x)?
3.How to find the derivative of e^x?
4.What is the derivative of ln(x)?
5.What is the derivative formula for a constant function?
Glossary for Derivative Formulas
- Derivative: A measure of how a function changes as its input changes, representing the slope of the function.
- Power rule: A basic rule to find the derivative of power functions, given by f'(x)=nx^(n-1).
- Chain rule: A method for calculating the derivative of composite functions.
- Product rule: A formula used to find the derivative of the product of two functions.
- Quotient rule: A formula used to find the derivative of the quotient of two functions.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
