Derivative Calculator

Definition and Fundamental Concept

The derivative of a function f(x) at a point x = a is defined as the limit of the difference quotient as h approaches zero. This limit represents the instantaneous rate of change and is the foundation of differential calculus.

f'(a) = \lim_h \to 0 (f(a+h) - f(a))/(h)

Basic Derivative Rules

Fundamental rules for finding derivatives: • Power Rule: d/dx(x^n) = n×x^(n-1) • Constant Rule: d/dx(c) = 0 for any constant c • Constant Multiple Rule: d/dx(cf(x)) = c×f'(x) • Sum Rule: d/dx(f(x) + g(x)) = f'(x) + g'(x) • Difference Rule: d/dx(f(x) - g(x)) = f'(x) - g'(x)

(d)/(dx)(x^n) = nx^n-1 \quad \text(Power Rule)

Constant and Linear Rules

Rules for constants and linear combinations:

(d)/(dx)(c) = 0 \quad \text(Constant Rule) \\ (d)/(dx)(cf(x)) = c(d)/(dx)f(x) \quad \text(Constant Multiple Rule) \\ (d)/(dx)[f(x) ± g(x)] = (d)/(dx)f(x) ± (d)/(dx)g(x) \quad \text(Sum/Difference Rule)

Product and Quotient Rules

For more complex functions involving multiplication and division: • Product Rule: d/dx(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) • Quotient Rule: d/dx(f(x)/g(x)) = (f'(x)g(x) - f(x)g'(x))/g(x)² These rules are essential for differentiating products and quotients of functions.

(d)/(dx)[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \quad \text(Product Rule) \\ (d)/(dx)[(f(x))/(g(x))] = (f'(x)g(x) - f(x)g'(x))/([g(x)]^2) \quad \text(Quotient Rule)

Chain Rule

The chain rule is crucial for composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). This rule allows us to differentiate complex nested functions by breaking them down into simpler components.

(d)/(dx)[f(g(x))] = f'(g(x)) · g'(x) \quad \text(Chain Rule) \\ (dy)/(dx) = (dy)/(du) · (du)/(dx) \quad \text(Alternative notation)

Trigonometric Derivatives

Derivatives of trigonometric functions: • d/dx(sin x) = cos x • d/dx(cos x) = -sin x • d/dx(tan x) = sec² x • d/dx(cot x) = -csc² x • d/dx(sec x) = sec x tan x • d/dx(csc x) = -csc x cot x

(d)/(dx)(\sin x) = \cos x \\ (d)/(dx)(\cos x) = -\sin x \\ (d)/(dx)(\tan x) = \sec^2 x \\ (d)/(dx)(\cot x) = -\csc^2 x \\ (d)/(dx)(\sec x) = \sec x \tan x \\ (d)/(dx)(\csc x) = -\csc x \cot x

Exponential and Logarithmic Derivatives

Derivatives involving exponentials and logarithms: • d/dx(e^x) = e^x • d/dx(a^x) = a^x ln(a) • d/dx(ln x) = 1/x • d/dx(log_a x) = 1/(x ln a) • d/dx(x^x) = x^x(1 + ln x)

(d)/(dx)(e^x) = e^x \\ (d)/(dx)(a^x) = a^x \ln a \\ (d)/(dx)(\ln x) = (1)/(x) \\ (d)/(dx)(\log_a x) = (1)/(x \ln a) \\ (d)/(dx)(x^x) = x^x(1 + \ln x)

Inverse Trigonometric Derivatives

Derivatives of inverse trigonometric functions: • d/dx(arcsin x) = 1/√(1-x²) • d/dx(arccos x) = -1/√(1-x²) • d/dx(arctan x) = 1/(1+x²) • d/dx(arccot x) = -1/(1+x²) • d/dx(arcsec x) = 1/(|x|√(x²-1)) • d/dx(arccsc x) = -1/(|x|√(x²-1))

(d)/(dx)(\arcsin x) = (1)/(√(1-x^2)) \\ (d)/(dx)(\arccos x) = -(1)/(√(1-x^2)) \\ (d)/(dx)(\arctan x) = (1)/(1+x^2) \\ (d)/(dx)(\textarccot x) = -(1)/(1+x^2) \\ (d)/(dx)(\textarcsec x) = (1)/(|x|√(x^2-1)) \\ (d)/(dx)(\textarccsc x) = -(1)/(|x|√(x^2-1))

Higher-Order Derivatives

Second and higher-order derivatives provide information about concavity and acceleration: • Second derivative: f''(x) = d/dx(f'(x)) • Third derivative: f'''(x) = d/dx(f''(x)) • nth derivative: f^(n)(x) = d/dx(f^(n-1)(x)) Higher derivatives are used in Taylor series and analyzing function behavior.

f''(x) = (d^2)/(dx^2)f(x) \\ f'''(x) = (d^3)/(dx^3)f(x) \\ f^(n)(x) = (d^n)/(dx^n)f(x) \\ \textNotation: f', f'', f''', f^(4), \ldots, f^(n)

Implicit Differentiation

When functions are defined implicitly (e.g., x² + y² = 1), we use implicit differentiation: • Differentiate both sides with respect to x • Treat y as a function of x • Solve for dy/dx This technique is essential for curves that cannot be expressed as y = f(x).

(d)/(dx)[x^2 + y^2] = 2x + 2y(dy)/(dx) = 0 \\ \textExample: x^2 + y^2 = 1 \Rightarrow 2x + 2y(dy)/(dx) = 0 \Rightarrow (dy)/(dx) = -(x)/(y)

Logarithmic Differentiation

For functions of the form f(x)^g(x) or complex products, use logarithmic differentiation: • Take natural logarithm of both sides • Differentiate implicitly • Solve for the derivative This technique simplifies complex differentiation problems.

\ln y = \ln f(x) \Rightarrow (y')/(y) = (f'(x))/(f(x)) \\ \textFor y = f(x)^g(x): \ln y = g(x)\ln f(x) \Rightarrow (y')/(y) = g'(x)\ln f(x) + g(x)(f'(x))/(f(x))

Applications in Physics

Derivatives are fundamental in physics: • Velocity: v(t) = ds/dt (derivative of position) • Acceleration: a(t) = dv/dt = d²s/dt² (derivative of velocity) • Force: F = ma = m(d²s/dt²) • Power: P = dW/dt (derivative of work) • Current: I = dQ/dt (derivative of charge)

v(t) = (ds)/(dt) \quad \text(Velocity) \\ a(t) = (dv)/(dt) = (d^2s)/(dt^2) \quad \text(Acceleration) \\ F = ma = m(d^2s)/(dt^2) \quad \text(Force) \\ P = (dW)/(dt) \quad \text(Power) \\ I = (dQ)/(dt) \quad \text(Current)

Applications in Economics

Derivatives are crucial in economic analysis: • Marginal cost: MC = dC/dq (derivative of cost function) • Marginal revenue: MR = dR/dq (derivative of revenue function) • Marginal utility: MU = dU/dx (derivative of utility function) • Elasticity: ε = (dq/dp) × (p/q) These concepts help optimize business decisions.

MC = (dC)/(dq) \quad \text(Marginal Cost) \\ MR = (dR)/(dq) \quad \text(Marginal Revenue) \\ MU = (dU)/(dx) \quad \text(Marginal Utility) \\ \varepsilon = (dq)/(dp) · (p)/(q) \quad \text(Elasticity)

Optimization Problems

Derivatives are essential for finding extrema: • Critical points: f'(x) = 0 or f'(x) undefined • First derivative test: determines if critical point is max/min • Second derivative test: f''(x) > 0 (concave up, minimum), f''(x) < 0 (concave down, maximum) • Global extrema: compare function values at critical points and endpoints

f'(x) = 0 \text or f'(x) \text undefined (Critical Points) \\ f''(x) > 0 \Rightarrow \textLocal Minimum \\ f''(x) < 0 \Rightarrow \textLocal Maximum \\ f''(x) = 0 \Rightarrow \textTest inconclusive (possible inflection point)

Related Rates

When two or more quantities change over time, related rates problems use the chain rule: • Identify the relationship between variables • Differentiate with respect to time • Substitute known values and solve for the unknown rate Common applications include expanding circles, filling containers, and moving objects.

(dA)/(dt) = (dA)/(dr) · (dr)/(dt) = 2π r (dr)/(dt) \\ \textExample: Circle area A = π r^2 \Rightarrow (dA)/(dt) = 2π r (dr)/(dt)

Hyperbolic Function Derivatives

Derivatives of hyperbolic functions: • d/dx(sinh x) = cosh x • d/dx(cosh x) = sinh x • d/dx(tanh x) = sech² x • d/dx(coth x) = -csch² x • d/dx(sech x) = -sech x tanh x • d/dx(csch x) = -csch x coth x

(d)/(dx)(\sinh x) = \cosh x \\ (d)/(dx)(\cosh x) = \sinh x \\ (d)/(dx)(\tanh x) = \textsech^2 x \\ (d)/(dx)(\coth x) = -\textcsch^2 x \\ (d)/(dx)(\textsech x) = -\textsech x \tanh x \\ (d)/(dx)(\textcsch x) = -\textcsch x \coth x

Worked Example

Inputs

• function: 3x^2 - 4x + 5
• functionType: Polynomial
• derivativeOrder: 1st

Result: 6x - 4

Explanation

The derivative of f(x) = 3x^2 - 4x + 5 with respect to x is f'(x) = 6x - 4. This is calculated using the power rule: d/dx(x^n) = n×x^(n-1).

Second Scenario

Inputs

• function: 3x^2 - 4x + 5
• functionType: Polynomial
• derivativeOrder: 1st

Result: 6x - 4

Explanation

This scenario uses different inputs (function = 3x^2 - 4x + 5, functionType = Polynomial, derivativeOrder = 1st) to show how changing one variable affects the derivative result. Run the calculator above with these values to get the exact updated output with step-by-step work.