Derivative Calculator – Differentiate Functions Step by Step
Calculate the derivative of any function with our free online derivative calculator. Applies power, product, quotient, and chain rules with detailed step-by-step differentiation shown.
Enter in format: ax^n (e.g., 3x^2, x^3, 5x, 7)
How the Derivative Calculator Works
This calculator applies the fundamental rules of differentiation to find the derivative of your function. The derivative represents the instantaneous rate of change of a function at any given point, which geometrically corresponds to the slope of the tangent line.
When you enter a function, the calculator identifies which differentiation rule applies:
- Power Rule: For functions like x^n, the derivative is n·x^(n-1)
- Product Rule: For u(x)·v(x), the derivative is u'v + uv'
- Quotient Rule: For u(x)/v(x), the derivative is (vu' - uv')/v²
- Chain Rule: For composite functions f(g(x)), the derivative is f'(g(x))·g'(x)
Example Calculations
Power Rule Example
Find the derivative of f(x) = 3x²
f(x) = 3x²
f'(x) = 3 · 2 · x^(2-1) = 6x
Product Rule Example
Find the derivative of f(x) = (2x+1)(x-3)
Let u = 2x+1, so u' = 2
Let v = x-3, so v' = 1
f'(x) = u'v + uv' = 2(x-3) + (2x+1)(1) = 2x - 6 + 2x + 1 = 4x - 5
Chain Rule Example
Find the derivative of f(x) = (3x+2)⁴
Outer function: u⁴, derivative: 4u³
Inner function: 3x+2, derivative: 3
f'(x) = 4(3x+2)³ · 3 = 12(3x+2)³
Quick Fact: The Birth of Calculus
The derivative was independently developed by both Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Newton called it the "method of fluxions" and used it to describe rates of change in physics, while Leibniz developed the notation dy/dx that we still use today. Their rivalry over who invented calculus first became one of the most famous disputes in the history of mathematics. Leibniz's notation proved more practical and is the standard we use in modern calculus.
Frequently Asked Questions
What is a derivative in calculus?
A derivative measures how a function changes as its input changes. It represents the instantaneous rate of change or the slope of the tangent line at any point on the function's graph. For example, if you have a function describing position over time, its derivative gives you velocity.
When do I use the product rule?
Use the product rule when differentiating two functions multiplied together, like f(x) = u(x)·v(x). The formula is: d/dx[u·v] = u'v + uv'. A common mistake is to think the derivative of a product is just the product of derivatives, which is incorrect.
What's the difference between the quotient and product rule?
The product rule handles multiplication: d/dx[u·v] = u'v + uv'. The quotient rule handles division: d/dx[u/v] = (vu' - uv')/v². Notice the quotient rule has subtraction in the numerator and the denominator squared, making it slightly more complex.
How do I know when to use the chain rule?
Use the chain rule for composite functions—when one function is "inside" another. Examples include (3x+2)⁴, sin(2x), or e^(x²). If you can identify an "outer" function and an "inner" function, you need the chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x).
What does the derivative tell me about a graph?
The derivative at a point gives the slope of the tangent line at that point. When f'(x) > 0, the function is increasing. When f'(x) < 0, the function is decreasing. When f'(x) = 0, you have a critical point that could be a maximum, minimum, or inflection point.
Can this calculator handle trigonometric functions?
Yes, the chain rule mode supports sin(ax+b) and cos(ax+b) functions. The derivatives follow standard rules: d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = -sin(x), combined with the chain rule for the inner function.
Why does the derivative of a constant equal zero?
A constant doesn't change, so its rate of change is zero. Geometrically, a constant function graphs as a horizontal line, which has a slope of 0. This is why d/dx[5] = 0 and why the derivative of any constant term disappears during differentiation.
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