Application of Derivatives Formula

Derivatives as the Rate of Change

(i) The average rate of change = Δy\over Δt

(ii) Instantaneous rate of change = \lim_{x \to 0} {Δy\over Δt} = dy\over dt

Marginal Function

(i) Cost Function {C(x)} = FC + V(x)

• Fixed Cost (FC)
• variable Cost {V(x)}

(ii) Revenue Function R(x) = p × x

• Cost per unit (p)
• Number of units (x)

(iii) Profit Function P(x) = R(x) – C(x)

(iv) Break-even point: R(x) – C(x) = 0

(v) Marginal Cost = dC(x)\over dx

(vi) Marginal Revenue = dR(x)\over dx

Tangents and Normal

Let y = f(x) be the curve and P (x₁, y₁) be the point at which tangent and normal are drawn then,

(i) Slope of the tangent (m₁) = dy\over dx

(ii) Slope of the tangent (m₂) = – 1\over {dy/dx}

(iii) Eq of tangent: y – y₁ = m₁(x – x₁)

(iv) Eq of Normal: y – y₁ = m₂(x – x₁)

(v) Length of Tangent = y₁ \sqrt{1+({dy\over dx})^2} \over {dy\over dx}

(vi) Length of the normal = y₁ \sqrt{1+({dy\over dx})^2}

(vii) Angle of the intersection of two curves

tan θ = |{m_1-m_2\over 1+m_1m_2}|

Rolle’s Theorem

If a function f(x) is

(i) continuous in the closed interval [a,b]

(ii) Derivable in the open interval (a,b) and

(iii) f(a) = f(b)

Then there exist at least one real number c in (a,b) such that f’(c) = 0

Lagrange’s Mean Value Theorem

If a function f(x) is

(i) continuous in the closed interval [a,b] and

(ii) Derivable in the open interval (a,b)

then there exists at least one real number c in (a,b) such that

Approximations and Errors

1. Let y = f(x) be a given function and Δx denotes a small increment in x, corresponding to which y increases by Δy. Then, for small increments, we assume that

∴ Δy = {dy\over dx}×Δx

For approximations of y, Δy = dy

Then, dy = {dy\over dx}×Δx

Thus y + Δy = f(x + Δx) = f(x) + {dy\over dx}×Δx

2. Let Δx be the error in the measurement of independent variable x and Δy is the corresponding error in the measurement of dependent variable y.

Then, Δy = {dy\over dx}×Δx

Δy =Absolute error in measurement of y