Methods of Differentiation

Differentiation is the process of finding the rate of change of a function with respect to its variable. The derivative of f(x) at x = a is defined as f'(a) = lim(h→0) [f(a+h) - f(a)]/h, provided this limit exists. Geometrically, f'(a) is the slope of the tangent to the curve y = f(x) at the point (a, f(a)).

Basic Differentiation Rules:

• Power Rule: d/dx(x^n) = n*x^(n-1), for any real n
• Constant Multiple: d/dx(cf) = cf'(x)
• Sum/Difference: d/dx(f +/- g) = f'(x) +/- g'(x)
• Product Rule: d/dx(f*g) = f'g + fg'
• Quotient Rule: d/dx(f/g) = (f'g - fg')/$g^{2}$, where g != 0
• Chain Rule: d/dx(f(g(x))) = f'(g(x)) * g'(x)

Standard Derivatives (must memorize):

• d/dx(sin x) = cos x, d/dx(cos x) = -sin x
• d/dx(tan x) = $sec^{2}$ x, d/dx(cot x) = -$cosec^{2}$ x
• d/dx(sec x) = sec x * tan x, d/dx(cosec x) = -cosec x * cot 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(sin^(-1) x) = 1/$\sqrt{1-x^{2}}$, d/dx(cos^(-1) x) = -1/$\sqrt{1-x^{2}}$
• d/dx(tan^(-1) x) = 1/(1+$x^{2}$), d/dx(cot^(-1) x) = -1/(1+$x^{2}$)
• d/dx(sec^(-1) x) = 1/(|x|$\sqrt{x^{2}-1}$), d/dx(cosec^(-1) x) = -1/(|x|$\sqrt{x^{2}-1}$)

Implicit Differentiation: When y is defined implicitly by F(x, y) = 0, differentiate both sides with respect to x, treating y as a function of x. Every term involving y gets multiplied by dy/dx via the chain rule.

Parametric Differentiation: If x = f(t) and y = g(t), then dy/dx = (dy/dt)/(dx/dt) = g'(t)/f'(t). The second derivative is $d^{2}$y/$dx^{2}$ = [d/dt(dy/dx)] / (dx/dt).

Logarithmic Differentiation: For functions of the form y = [f(x)]^g(x) or products/quotients of many functions, take ln of both sides first: ln y = g(x)*ln(f(x)), then differentiate implicitly.

Higher Order Derivatives: The second derivative f''(x) = d/dx(f'(x)), the third derivative f'''(x) = d/dx(f''(x)), and so on. Leibniz's Theorem for the nth derivative of a product: (uv)^(n) = sum(r=0 to n) C(n,r) * u^(n-r) * v^(r).

Differentiation of Determinants: If a determinant has rows/columns as functions of x, differentiate by differentiating one row (or column) at a time while keeping others unchanged, then summing all such determinants.

The key problem-solving concept is recognizing which differentiation technique (chain rule, implicit, parametric, logarithmic, or substitution) most efficiently simplifies the given expression, particularly when inverse trigonometric functions can be simplified using appropriate substitutions before differentiating.