Derivative Calculator
A derivative calculator finds the derivative of a mathematical function with respect to a variable. Derivatives are used in calculus to measure rates of change, slopes, motion, optimization, physics, engineering, and many other mathematical applications.
What Is a Derivative?
A derivative describes how quickly a function changes as its input changes. Geometrically, the derivative at a point represents the slope of the tangent line to the function at that point.
Basic Derivative Formula
The derivative of a function f(x) can be defined using the limit:
fβ²(x) = limh β 0 [f(x + h) β f(x)] Γ· h
This definition describes the instantaneous rate of change of the function.
Basic Power Rule
One of the most commonly used derivative rules is the power rule:
d/dx (xn) = nxnβ1
For example:
d/dx (x3) = 3x2
Therefore, the derivative of x3 is 3x2.
Common Derivative Rules
| Function | Derivative |
|---|---|
| c | 0 |
| xn | nxnβ1 |
| sin(x) | cos(x) |
| cos(x) | βsin(x) |
| ex | ex |
| ln(x) | 1/x |
Sum Rule
The derivative of a sum is the sum of the individual derivatives:
(f + g)β² = fβ² + gβ²
For example:
d/dx (x2 + 3x) = 2x + 3
Product Rule
When two functions are multiplied, the product rule is used:
(fg)β² = fβ²g + fgβ²
This rule is useful when a function cannot be differentiated by treating the factors independently.
Why Are Derivatives Useful?
Derivatives can identify how quantities change, find slopes of curves, locate maximum and minimum values, and describe motion. For example, the derivative of position with respect to time gives velocity.
Common Mistakes to Avoid
Make sure the variable being differentiated is clear. When using the power rule, reduce the exponent by one after multiplying the original exponent by the coefficient.
For functions involving products, quotients, or compositions, use the appropriate differentiation rule rather than applying the power rule alone.
Quick Summary
A derivative measures the instantaneous rate of change of a function and represents the slope of its tangent line. Common rules include the power rule, sum rule, product rule, and derivatives of standard functions such as sine, cosine, exponential, and logarithmic functions.