Derivadas Calculator
Enter a function of x, then calculate the symbolic derivative and the slope at a specific point.
- Use operators: +, -, *, /, ^
- Functions: sin, cos, tan, exp, log, sqrt
- Example formats: x^2 + 3*x - 1 sin(x)^2 exp(2*x)
What Is a Derivative?
In calculus, a derivative measures how fast a function changes at any specific point. If a function describes position, the derivative describes velocity. If a function describes cost, the derivative can describe marginal cost. In geometric terms, the derivative gives the slope of the tangent line to a curve.
A derivadas calculator helps you skip repetitive algebra and focus on interpretation. Instead of spending all your energy on mechanical steps, you can spend time understanding behavior: where a function increases, where it decreases, and how quickly those changes happen.
How to Use This Derivadas Calculator
1) Enter the function
Type your function in the f(x) field using standard math syntax. Include multiplication signs where needed, such as 3*x rather than 3x.
2) Enter a point x
If you want the numeric derivative at a specific point, add a value in the x field. The calculator will show:
- Function value f(x)
- First derivative f′(x)
- Second derivative approximation f″(x)
- Tangent line equation at that point
3) Adjust the step size h if needed
The value h controls numerical precision for finite-difference methods. Smaller values often improve accuracy, but extremely tiny values may increase floating-point error. The default of 0.0001 works well for most smooth functions.
Supported Syntax and Function Types
This calculator handles a broad set of expressions used in typical algebra and calculus courses:
- Polynomials: x^5 - 2*x + 9
- Rational functions: (x^2 + 1)/(x - 3)
- Trigonometric: sin(x), cos(x), tan(x)
- Exponential and logarithmic: exp(x), log(x)
- Compositions: sin(x^2), exp(sin(x))
Quick Derivative Rule Refresher
Power Rule
If f(x) = x^n, then f′(x) = n*x^(n-1).
Product Rule
If f(x) = u(x)v(x), then f′(x) = u′(x)v(x) + u(x)v′(x).
Quotient Rule
If f(x) = u(x)/v(x), then f′(x) = (u′v - uv′)/v^2.
Chain Rule
If f(x)=g(h(x)), then f′(x)=g′(h(x))*h′(x).
Why This Tool Shows Both Symbolic and Numeric Results
Symbolic differentiation gives an exact expression, which is useful for proofs, simplification, and repeated evaluation. Numerical differentiation gives an approximation at a point and can still work when symbolic forms become messy. Seeing both methods helps build conceptual understanding and practical confidence.
Common Mistakes to Avoid
- Forgetting multiplication signs (write 2*x, not 2x).
- Using invalid domains (for example, log(x) requires x > 0).
- Choosing an x-value where the function is undefined (like dividing by zero).
- Entering an h value that is zero or negative.
Applications in Real Life
Derivatives power optimization, machine learning, economics, physics, and engineering. They help estimate growth rates, find minima and maxima, model acceleration, and understand sensitivity. Even if you are just beginning calculus, mastering derivatives opens the door to higher-level quantitative thinking.
Final Takeaway
A good derivadas calculator is more than a shortcut. It is a study companion that lets you test hypotheses, verify hand calculations, and build intuition around slopes and rates of change. Use this page to practice often: change functions, try different points, and compare results until derivative behavior feels natural.