Derivative Calculator
Differentiate any function symbolically (first, second or third order) and evaluate the derivative at a point.
Details
Use ^ for powers, and sin, cos, ln or sqrt for functions.
You are differentiating
d/dx ( x³ + 2x )
1st derivative
…
This differentiates a function and shows how. Type the expression, say which variable you are differentiating with respect to, and it returns the derivative.
Ask for a second or third derivative if you need one, and enter a value for x to get the exact slope at that point rather than the general rule.
What is a derivative?
A derivative measures how fast something is changing at one exact moment.
Speed is the everyday example. Driving 60 miles in 2 hours averages 30 mph, but your speedometer does not show the average, it shows how fast you are going *right now*. The derivative is the speedometer reading.
On a graph it is the slope of the curve at a single point. A straight line has one slope everywhere, which is easy. A curve keeps changing direction, so its slope depends on where you stand on it, and the derivative is the rule that gives you the slope anywhere.
d/dx is an instruction, not a fraction: it says 'differentiate this with respect to x'. You may also see it written f′(x), said 'f prime of x'. Both mean the same thing.
What to enter
- Your function
- Type it as you would say it: x^2 for x squared, 3x for three x, sin(x), e^x, ln(x). Use ^ for powers and * between multiplied terms if it is ambiguous.
- Differentiate with respect to
- Which letter is the variable. Usually x, but t is common for anything changing over time.
- How many times?
- Differentiating twice gives the second derivative, which measures how fast the rate itself is changing. In motion, the first derivative is speed and the second is acceleration.
- At a point (optional)
- Leave blank for the general rule. Enter a number to get the exact slope at that spot.
The rules you will use most
- Power rule
- d/dx of xⁿ = n·xⁿ⁻¹. Bring the power down in front, then knock one off it. This covers most homework.
- Constant rule
- d/dx of any plain number = 0. A constant never changes, so its rate of change is nothing.
- Sum rule
- Differentiate each term separately and add the results.
- Product rule
- For two things multiplied: (first × derivative of second) + (second × derivative of first).
- Chain rule
- For a function inside another: differentiate the outside, then multiply by the derivative of the inside.
What this assumes
Angles in trigonometric functions are in radians, which is standard in calculus. The derivative of sin(x) is only cos(x) in radians.
The function is smooth at the point you ask about. A sharp corner has no single slope there.
How to calculate a derivative
Almost every polynomial derivative comes from one rule, and it is quicker than it looks.
- n
- The power on x
- n · x
- Multiply by the old power, then reduce the power by one
Take one term at a time. Terms added together can be differentiated separately, so break the function up first.
Bring the power down. For x³, the 3 comes down in front.
Knock one off the power. x³ becomes 3x². Any plain number on its own disappears entirely.
Substitute if you want a value. The derivative is a rule. Put a number into it to get the slope at that particular point.
See a worked example: the slope of x² at the point x = 3
- Function
- x²
- At
- x = 3
Power rule: bring the 2 down in front, then reduce the power from 2 to 1.
So the derivative is 2x¹, which is just 2x.
That is the rule for the slope anywhere on the curve.
Now put in x = 3: 2 × 3 = 6. So the curve is climbing at a slope of 6 at that exact point.
d/dx (x²) = 2x, and at x = 3 the slope is 6
Frequently asked questions
For a straight line they are the same thing, and the slope calculator handles that case with just two points.
A curve has a different slope at every point, so you cannot use two points. The derivative gives you a formula that produces the correct slope wherever you ask.
It means 'how much y changes for a tiny change in x'. It looks like a fraction because it came from one: the rise over run of two points squeezed infinitely close together.
Treat it as a single symbol rather than a division. f′(x) means exactly the same and is quicker to write.
Because a constant does not change. Graph y = 5 and you get a flat horizontal line, and a flat line has a slope of zero.
This is why +7 vanishes when you differentiate x² + 7. It also explains why the reverse operation, integration, has to add '+ C' back on: that information was destroyed.
It is the rate at which the rate is changing. With position, the first derivative is velocity and the second is acceleration.
On a graph it tells you which way the curve bends: positive curves upward like a cup, negative curves downward like a dome. That is how you tell a maximum from a minimum.
Whenever one function sits inside another, like sin(3x) or (2x + 1)⁵. The giveaway is a bracket with something more complicated than a single letter inside.
Differentiate the outside first, leaving the inside alone, then multiply by the derivative of the inside. For (2x + 1)⁵ that gives 5(2x + 1)⁴ × 2.
Problems people actually run into
Forgetting the chain rule on anything inside a bracket
Differentiating sin(3x) as cos(3x) is the single most common calculus error. The correct answer is 3cos(3x), because the inside also has to be differentiated and multiplied on.
A quick habit that catches it: whenever you see a bracket, ask 'is there anything more than a plain x inside?'. If yes, the chain rule applies.
Reading the answer as a number when it is a rule
The derivative of x² is 2x, which is still a formula, not a value. It only becomes a number once you choose a point.
Exam questions often ask for both, so read carefully: 'find the derivative' wants 2x, while 'find the gradient at x = 3' wants 6.
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Last updated: September 3, 2026