Derivative Calculator — Find f'(x) Step by Step
Calculate derivatives of polynomial, trigonometric, exponential, and logarithmic functions with step-by-step solution. Evaluate the derivative at any point. See also our Slope Calculator and Scientific Calculator.
How to Calculate Derivatives
- Select the function type (polynomial, trig, exponential, or logarithmic).
- Enter the coefficients or parameters for your function.
- Optionally enter a point x to evaluate the derivative at.
- Click Calculate to see the derivative and step-by-step solution.
Formula
Power Rule: d/dx[xⁿ] = n·xⁿ⁻¹
Constant Multiple: d/dx[c·f(x)] = c·f'(x)
Sum Rule: d/dx[f(x) + g(x)] = f'(x) + g'(x)
Trig Derivatives:
d/dx[sin(x)] = cos(x)
d/dx[cos(x)] = -sin(x)
d/dx[tan(x)] = sec²(x)
Exponential: d/dx[e^x] = e^x, d/dx[a^x] = a^x·ln(a)
Logarithmic: d/dx[ln(x)] = 1/x, d/dx[log_a(x)] = 1/(x·ln(a))Example
Find the derivative of f(x) = 3x² + 2x - 5:
d/dx[3x²] = 6x (power rule)
d/dx[2x] = 2 (power rule)
d/dx[-5] = 0 (constant rule)
f'(x) = 6x + 2
At x = 2: f'(2) = 6(2) + 2 = 14
Common Derivatives Reference Table
| f(x) | f'(x) | Rule |
|---|---|---|
| xⁿ | n·xⁿ⁻¹ | Power Rule |
| sin(x) | cos(x) | Trig |
| cos(x) | -sin(x) | Trig |
| tan(x) | sec²(x) | Trig |
| e^x | e^x | Exponential |
| a^x | a^x·ln(a) | Exponential |
| ln(x) | 1/x | Logarithmic |
| log_a(x) | 1/(x·ln(a)) | Logarithmic |
Frequently Asked Questions
What is a derivative?
A derivative measures the instantaneous rate of change of a function. Geometrically, it gives the slope of the tangent line at any point on the curve.
What is the power rule?
The power rule states that d/dx[xⁿ] = n·xⁿ⁻¹. Bring the exponent down as a coefficient and reduce the exponent by 1.
What does the derivative at a point mean?
The derivative at a specific point gives the slope of the function at that point. A positive value means the function is increasing; negative means decreasing.
Can this calculator handle chain rule?
This calculator handles basic function types. For composite functions requiring the chain rule, break them into simpler parts or use the product/quotient rules manually.
What is the difference between derivative and integral?
The derivative finds the rate of change (slope). The integral finds the accumulated area under a curve. They are inverse operations by the Fundamental Theorem of Calculus.