Derivative Calculator — Differentiation with Step-by-Step
Calculate derivatives of polynomial, trigonometric, and exponential functions. Enter your function type and coefficients to get the derivative with step-by-step rules applied.
Original Function f(x)
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Derivative f'(x)
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Step-by-Step Solution
Differentiation Rules Reference
- Power Rule: d/dx[axⁿ] = a·n·xⁿ⁻¹
- Constant Rule: d/dx[c] = 0
- Sin Rule: d/dx[sin(x)] = cos(x)
- Cos Rule: d/dx[cos(x)] = -sin(x)
- Exponential Rule: d/dx[eˣ] = eˣ
- Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
- Ln Rule: d/dx[ln(x)] = 1/x
Frequently Asked Questions
What is a derivative?
A derivative measures the rate of change of a function with respect to its variable. For f(x), the derivative f'(x) gives the slope of the tangent line at any point x. It represents instantaneous rate of change.
What is the power rule?
The power rule states that d/dx[xⁿ] = n·xⁿ⁻¹. For example, d/dx[x³] = 3x². If there is a coefficient a, then d/dx[axⁿ] = a·n·xⁿ⁻¹.
What are the derivatives of trig functions?
d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x), d/dx[tan(x)] = sec²(x). These are fundamental results used throughout calculus.
What is the derivative of eˣ?
The derivative of eˣ is eˣ itself — it is unchanged by differentiation. More generally, d/dx[eᵃˣ] = a·eᵃˣ using the chain rule.
What is the chain rule?
The chain rule is used when differentiating composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). For example, d/dx[sin(2x)] = cos(2x) · 2 = 2cos(2x).