Differentiability and Differentiation

• The rate of change of a quantity y with respect to another quantity x is called the derivative of y with respect to x.

• If a function is not derivable at a point, it need not imply that it is discontinuous at that point. But, however, discontinuity at a point necessarily implies non-derivability.

• In case, a function is not differentiable but is continuous at a particular point say x = a, then it geometrically implies a sharp corner at x = a.

• A function f is said to be derivable over a closed interval [a, b] if :

• For the points a and b, f'(a+) and f'(b-) exist and

• For ant point c such that a < c < b, f'(c⁺) and f'(c^-) exist and are equal.

• If y = f(u) and u = g(x), then dy/dx = dy/du.du/dx = f'(g(x)) g'(x). This method is also termed as the chain rule.

• For composite functions, differentiation is carried out in this way:

    If y = [f(x)]ⁿ, then we put u = f(x). So that y = uⁿ. Then by chain rule:

    dy/dx = dy/du.du/dx = nu^(n-1)f' (x) = [f(x)]^(n-1) f' (x)

• If the functions f(x) and g(x) are derivable at x = a, then the following functions are also derivable:

• f(x) + g(x)

• f(x) - g(x)

• f(x) . g(x)

• f(x) / g(x), provided g(a) ≠ 0

• If the function f(x) is differentiable at x = a while g(x) is not derivable at x = a, then the product function f(x). g(x) can still be differentiable at x = a.

• Even if both the functions f(x) and g(x) are not differentiable at x = a, the product function f(x).g(x) can still be differentiable at x = a.

• Even if both the functions f(x) and g(x) are not derivable at x = a, the sum function f(x) + g(x) can still be differentiable at x = a. 

• If function f(x) is derivable at x = a, this need not imply that f'(x) is continuous at x = a.

• Differentiation using substitution: The following substitutions may be used for computing the differentiation of the functions:

    √a² - x², use x = a sin θ or a cos θ

    √a² + x², use x = a tan θ or a cot θ

    √x² - a², use x = a sec θ or a cosec θ

    √(a + x) or √(a – x) , use x = a cos 2θ

    √2ax - x², use x = a (1 – cos θ)

• If x = f(t) and y = g(t), where t is a parameter, then dy/dx = (dy/dt)/(dx/dt) = g’(t)/f’(t)

• If a function is in the form of exponent of a function over another function such as f(x)^g(x), we first take logarithm and then differentiate.

• Let y = f(x) and z = g(x), then the differentiation of y with respect to z is dy/dz = dy/dx / dz/dx = f’(x)/g’(x)

• Leibnitz Theorem:

    If u and v are two functions of x possessing nth order derivatives, then

    Dⁿ(u.v) = ⁿC₀(Dⁿu)v + ⁿC₁(D^n-1u)(Dv) + ⁿC₂(D^n-2u)(D²v) + ….….. ….…. + ⁿC_r(D^n-ru)(D^rv) + ….…. + ⁿC_n(Dⁿv)

• Partial coefficient of f(x,y) with respect to x is the ordinary differential coeffiicient of f(x, y) when y is regarded as a constant.

• If f”(x, y) be a function of two variables such that ∂f/∂x and ∂f/∂y both exist then we can talk of second order partial derivatives as well.

    The partial derivative of ∂f/∂x with respect to x is f_xx i.e. ∂²f/∂x²

    The partial derivative of ∂f/∂x with respect to y is f_xy i.e. ∂²f/∂x∂y

    The partial derivative of ∂f/∂y with respect to x is f_yx i.e. ∂²f/∂y∂x

    The partial derivative of ∂f/∂y with respect to y is f_yy i.e. ∂²f/∂y²

    Note: ∂²f/∂x∂y = ∂²f/∂y∂x.

• Let f(x, y) be a homogeneous function of order n so that f(tx, ty) = tⁿf(x, y), then

• x_i ∂f/∂x_i = n f(x)

• x²∂²f/∂x² + 2xy ∂²f/∂x∂y + y²∂²f/∂y² = n(n-1)f(x,y)