Derivative Calculator with Steps
Stuck on a calculus problem? Our tool not only gives you the answer but also shows you the step-by-step logic, helping you understand the 'why' behind the math. Master differentiation, one problem at a time.
The 'Aha!' Moment: A Student's Story
Meet Maya. To her, calculus felt like a list of abstract rules. She could solve for `f'(x)`, but she kept wondering, "What am I actually finding?"
One day, while watching a rocket launch video, she asked her teacher, "We can find the rocket's average speed, but how do we know its exact speed at the 3-second mark?"
Her teacher smiled. "That's exactly what a derivative is," she said. "It's not the average rate of change over time; it's the instantaneous rate of change at one precise moment."
Suddenly, it clicked. Derivatives weren't just about symbols on a page. They were about finding the "right now" speed of a rocket, the "at this very second" growth of a company's profit, or the exact slope of a roller coaster track at its steepest point. The abstract rule became a powerful tool for understanding the world in motion.
From Skateboards to Wall Street: Where Derivatives Show Up
Physics & Motion
Ever wonder how a self-driving car knows its exact speed at any given moment? Velocity is the derivative of the position function. Derivatives are fundamental to describing motion, acceleration, and forces.
Economics & Business
Companies use derivatives to find the "marginal cost"—the exact cost of producing just one more item. This helps them make critical decisions about pricing and production levels to maximize profit.
Video Games & Animation
To create smooth, realistic character movements, game developers use derivatives to calculate how a character's position should change from one frame to the next. It's the math behind lifelike animation!
Engineering & Optimization
Engineers use derivatives to find the maximum or minimum values of a function. This is crucial for optimizing everything from the shape of an airplane wing for maximum lift to the size of a soda can to hold the most liquid with the least material.
Understanding the Rules of Differentiation
The Power Rule
The Power Rule is the most fundamental rule of differentiation. It's used for terms in the form `ax^n`.
Rule: `d/dx (x^n) = n * x^(n-1)`
Example: The derivative of `x^3` is `3x^2`.
The Sum/Difference Rule
This rule states that you can differentiate a function term by term. The derivative of a sum is the sum of the derivatives.
Rule: `d/dx (f(x) + g(x)) = f'(x) + g'(x)`
The Constant Rule
The derivative of any constant number is always zero. This is because a constant doesn't change, so its rate of change is 0.
Rule: `d/dx (c) = 0`
Example: The derivative of `7` is `0`.
Frequently Asked Questions
What is a derivative?
In simple terms, the derivative of a function measures the sensitivity to change of the function's value with respect to a change in its argument. For a function graphed on a 2D plane, the derivative at a point is the slope of the tangent line to the curve at that point.
What does f'(x) mean?
f'(x), read as "f prime of x", is one of the most common notations for the derivative of a function f(x). Another common notation is `d/dx`, which means "the derivative with respect to x".
How do you find the derivative of x^2?
Using the Power Rule, where n=2. The rule is `d/dx (x^n) = n * x^(n-1)`. So, for x², you bring the power (2) down as a multiplier and subtract one from the power: `2 * x^(2-1)`, which simplifies to `2x`.
What functions does this calculator support?
Currently, this calculator is designed to differentiate polynomial functions (e.g., `4x^3 - x^2 + 5`). It supports the power rule, constant multiple rule, sum rule, and constant rule. Support for more complex functions like trigonometric, exponential, and logarithmic functions will be added in the future.