Derivative Calculator
Find the derivative of a function of x with the rules used, the first, second or third derivative, the slope at a point and the tangent line.
Results
| Item | Value |
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Steps
| Step | Working |
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Quick Answer
The derivative tells you how fast a function changes, or the slope of its graph at each point. Using the power rule, the derivative of f(x) = x^3 + 2x^2 − 5x + 1 is f′(x) = 3x^2 + 4x − 5. At x = 2 the slope is 15, so the tangent line there is y = 15x − 23.
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Table of Contents
- Derivative Calculator
- Quick Answer
- Related Calculators
- How This Derivative Calculator Works
- How to Use the Derivative Calculator
- Core Derivative Rules
- Key Terms
- Worked Example
- Real-Life Examples
- Derivative vs Integral
- Common Derivatives
- Tips for Typing Functions
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Derivative Calculator Works
Type a function of x, such as x^2 sin(x) or ln(x^2 + 1), and choose the first, second or third derivative. The calculator reads the function and applies the rules of differentiation, such as the power, product, quotient and chain rules, then simplifies the result. If you enter a point, it also gives the slope there, the tangent line, and a numeric check of the slope.
Quick explanation: the derivative f′(x) is the slope of f(x) at each point.
- Positive f′(x): the function is going up.
- Negative f′(x): the function is going down.
- f′(x) = 0: a possible peak, valley or flat point.
How to Use the Derivative Calculator
- Type your function using x as the variable.
- Choose the first, second or third derivative.
- Optionally enter an x value to find the slope at that point.
- Press Calculate.
- Read the derivative, the rules used, and the slope and tangent line.
Core Derivative Rules
- Power rule: (xn)′ = n·xn−1
- Constant multiple: (c·f)′ = c·f′
- Sum rule: (f + g)′ = f′ + g′
- Product rule: (f·g)′ = f′·g + f·g′
- Quotient rule: (f / g)′ = (f′·g − f·g′) / g2
- Chain rule: f(g(x))′ = f′(g(x))·g′(x)
Key Terms
- Derivative: the rate of change of a function, written f′(x) or dy/dx.
- Tangent line: a straight line that touches a curve at a point with the same slope.
- Second derivative: the derivative of the derivative, showing how the slope changes.
- Chain rule: the rule for differentiating a function inside another function.
- Critical point: a point where the derivative is zero or undefined.
Worked Example
Find the derivative of f(x) = sin(x2).
- The outer function is sin(u) with u = x2, so use the chain rule.
- (sin u)′ = cos(u)·u′, and u′ = 2x by the power rule.
- f′(x) = cos(x2)·2x = 2x·cos(x2)
Real-Life Examples
- Homework: check derivatives step by step, such as (x2 + 1)3, whose derivative is 6x·(x2 + 1)2.
- Peaks and valleys: f(x) = x3 − 3x has f′(x) = 3x2 − 3, which is zero at x = −1 and x = 1.
- Speed: if position is s(t) = 5t2, speed is s′(t) = 10t. Type t as x.
- Concavity: the second derivative of x4 is 12x2, which shows how the curve bends.
Derivative vs Integral
- Derivative: the rate of change, or slope, of a function.
- Integral: the accumulated total, or area under the curve.
- Link: by the fundamental theorem of calculus, they undo each other.
Common Derivatives
| Function | Derivative |
|---|---|
| c (a constant) | 0 |
| xn | n·xn−1 |
| ex | ex |
| ax | ax·ln(a) |
| ln(x) | 1 / x |
| log(x), base 10 | 1 / (x·ln(10)) |
| √x | 1 / (2√x) |
| sin(x) | cos(x) |
| cos(x) | −sin(x) |
| tan(x) | sec(x)2 |
| sec(x) | sec(x)·tan(x) |
| csc(x) | −csc(x)·cot(x) |
| cot(x) | −csc(x)2 |
| arcsin(x) | 1 / √(1 − x2) |
| arccos(x) | −1 / √(1 − x2) |
| arctan(x) | 1 / (1 + x2) |
Tips for Typing Functions
- Powers: type x^2, or x^(1/2) for a square root.
- Multiplying: 3x and 3*x both work. x sin(x) means x times sin(x).
- Brackets: use them for function inputs: sin(2x), not sin 2x.
- Constants: e and pi are known numbers, not variables.
Common Mistakes to Avoid
- Forgetting the chain rule. The derivative of sin(3x) is 3cos(3x), not cos(3x).
- Multiplying derivatives for a product. (f·g)′ is not f′·g′. Use the product rule.
- Mixing up the quotient rule order. It is f′g − fg′ on top, not fg′ − f′g.
- Treating e as a variable. e is a constant, so the derivative of e2 is 0.
- Using degrees. Calculus formulas for sin and cos use radians.
Frequently Asked Questions
The instant rate of change of a function, or the slope of its graph at a point.
Apply rules such as the power, product, quotient and chain rules, or type the function here.
2x, by the power rule.
e^x. It is its own derivative.
1 / x, for x greater than zero.
cos(x). The derivative of cos(x) is −sin(x).
It shows concavity and helps tell whether a critical point is a peak or a valley.
Find f′(x), then put in the x value. Enter the point in the optional box here.
With slope m = f′(a) and point (a, f(a)), the line is y = f(a) + m(x − a).
Yes. Trig functions use radians, as in calculus.
Derivatives can be written in many equal forms. Check by entering a point and comparing the values.
It estimates the slope from two nearby points, (f(a + h) − f(a − h)) / 2h, so you can confirm the symbolic answer.
No. It finds ordinary derivatives of functions of x only.
References & Sources
| Source | Used For | Link |
|---|---|---|
| OpenStax, Calculus Volume 1, Chapter 3: Derivatives | Free textbook on derivative rules, the chain rule and tangent lines. | https://openstax.org/books/calculus-volume-1/pages/3-introduction |
| Wikipedia, Derivative | Definition, notation and rules of differentiation. | https://en.wikipedia.org/wiki/Derivative |