Differentiation

Derivative

• Derivative represents the rate of change of one quantity with respect to another.
• If y = f(x), then:
• dy/dx = lim_Δx→0 [f(x + Δx) − f(x)] / Δx
• Also denoted by: y′, D_xy, f′(x)
• The process of finding derivative is called differentiation.

Derivatives of Standard Functions

• d/dx (constant) = 0
• d/dx (xⁿ) = n xⁿ⁻¹
• d/dx (eˣ) = eˣ
• d/dx (aˣ) = aˣ ln a
• d/dx (ln x) = 1/x
• d/dx (logₐ x) = 1/(x ln a)
• d/dx (sin x) = cos x
• d/dx (cos x) = −sin x
• d/dx (tan x) = sec²x
• d/dx (cot x) = −cosec²x
• d/dx (sec x) = sec x tan x
• d/dx (cosec x) = −cosec x cot x

Derivatives of Inverse Trigonometric Functions

• d/dx (sin⁻¹x) = 1/√(1 − x²), −1 < x < 1
• d/dx (cos⁻¹x) = −1/√(1 − x²), −1 < x < 1
• d/dx (tan⁻¹x) = 1/(1 + x²)
• d/dx (cot⁻¹x) = −1/(1 + x²)
• d/dx (sec⁻¹x) = 1/(|x|√(x² − 1)), |x| > 1
• d/dx (cosec⁻¹x) = −1/(|x|√(x² − 1)), |x| > 1

Important Differentiation Formulae

• d/dx (e^ax sin bx) = e^ax(a sin bx + b cos bx)
• d/dx (e^ax cos bx) = e^ax(a cos bx − b sin bx)
• d/dx |x| = x/|x|, x ≠ 0
• d/dx (log |x|) = 1/x, x ≠ 0

Rules of Differentiation

• Sum/Difference: d/dx [f(x) ± g(x)] = f′(x) ± g′(x)
• Constant multiple: d/dx [k f(x)] = k f′(x)
• Product rule: d/dx [f(x)g(x)] = f′(x)g(x) + f(x)g′(x)
• Quotient rule: d/dx [f(x)/g(x)] = [g(x)f′(x) − f(x)g′(x)] / [g(x)]²

Chain Rule

• If y = f(t) and t = g(x), then:
• dy/dx = (dy/dt)(dt/dx)
• d/dx [g(f(x))] = g′(f(x)) · f′(x)

Parametric Differentiation

• If x = φ(t), y = ψ(t), then:
• dy/dx = (dy/dt)/(dx/dt)

Differentiation w.r.t. Another Function

• If f(x) and g(x) are functions of x:
• d(f(x))/d(g(x)) = (df/dx)/(dg/dx)

Implicit Differentiation

• If y is not expressed explicitly in terms of x:
• Differentiate both sides w.r.t. x treating y as a function of x
• Collect all dy/dx terms on one side

Short Method (Implicit Function)

• If f(x, y) = constant, then:
• dy/dx = −(∂f/∂x)/(∂f/∂y)

Logarithmic Differentiation

• Used when function is of the form:
• [f(x)]^g(x)
• Or product of three or more functions
• Take log on both sides before differentiating

Useful Substitutions

• If expression contains √(a² − x²) → put x = a sinθ or a cosθ
• If expression contains √(x² + a²) → put x = a tanθ or a cotθ
• If expression contains √(x² − a²) → put x = a secθ or a cosecθ
• If expression contains √((a − x)/(a + x)) → put x = a cosθ

Differentiation of a Determinant

• Differentiate determinant by differentiating one row at a time
• Sum of all such determinants gives the derivative
• Applicable when each element is a function of x

JEE Main Focus Tips

• Memorise standard derivatives thoroughly
• Chain rule and implicit differentiation are high-frequency
• Be careful with modulus and inverse trigonometric functions
• Check domain restrictions before applying formulas