Compute derivative formulas with step-by-step differentiation
Select Function Type
f(x) = a*x^n
f(x)=x^
f(x) = sin(ax)
f(x)=sin(x)
f(x) = e^(ax)
f(x)=e^(x)
f(x) = ln(ax)
f(x)=ln(x)
Result
f(x)
Derivative f(x)
Derivation
Derivative Rules
Power rule: d/dx(x^n) = n*x^(n-1)
Trig: d/dx sin(x) = cos(x), d/dx cos(x) = -sin(x)
Exp: d/dx e^x = e^x, Log: d/dx ln(x) = 1/x
Chain rule: d/dx f(g(x)) = f(g(x)) * g(x)
The derivative measures the instantaneous rate of change of a function. It is the foundation of calculus, with applications in physics (velocity), economics (marginal analysis), and optimization.
⚠The derivative at a point equals the slope of the tangent line at that point. If derivative = 0, the point is a critical point.
What Is a Derivative?
The derivative f(x) = lim h->0 (f(x+h)-f(x))/h. It represents the slope of the tangent line. Key rules include power rule, product rule, quotient rule, and chain rule. Each function type has specific differentiation formulas.
Power Rule
d/dx(x^n) = n*x^(n-1). Works for any real n. Example: x^3 -> 3x^2, 1/x = x^(-1) -> -x^(-2) = -1/x^2.
Chain Rule
d/dx f(g(x)) = f(g(x)) * g(x). Differentiate outer, then multiply by inner derivative. Used for all composite functions.
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