Leibnitz’s Rule Leibnitz’s Rule: If f is a continuous function on [a, b], and u(x) and v(x) are differentiable function of x whose values lie in [a, b], then . Proof:  Let d/dx(F(x)) = f(x) , , , . Example: If  (where x > 0), then find dy/dx. Solution: , , , , , , Read more about Leibnitz’s Rule[…]

Limits of the Form – 1.   , 2. . If Then  , , , . Example: Evaluate  Solution: , , , , , , , , . Share Tweet View Email Print Follow

Limits of the Form – Part 1 Limits of the Form – Form: : Let  then Taking log on both sides , , , , . Example: Evaluate  Solution: Given that Let us consider  Taking log on both sides , , , , . Share Tweet View Email Print Follow

Trajectories Trajectories:  We are given the family of plane curve F (x, y, a) = 0 depending on a single parameter a.  A curve making at each of its points a fixed angle α with the curve of the family passing through that point is called as isogonal trajectory of that family. If in particular Read more about Trajectories[…]

Geometrical Applications of Differential Equation Geometrical Applications of Differential Equation: We also use differential equations for finding the family of curves for which some conditions involving the derivatives are given. For this we proceed in the following way Equation of the tangent at a point (x, y) to the curve y = f(x) is given Read more about Geometrical Applications of Differential Equation[…]

Differentiation of Determinants To differentiation a determinant, we differentiate one row (or column) at a time, keeping other unchanged. For example, if . Differentiation on both sides, . (or) , Similar results hold for the differentiation of determinants of higher order. Example: If , then prove that f’(x) = 3x² + 2x (a² + b² Read more about Differentiation of Determinants[…]

Differentiation Using Logarithm If y = [f₁(x)]f₂(x) (or) y = f₁(x) f₂(x) f₃(x) … (or) . (i) y = [f₁(x)]f₂(x) log y = log [f₁(x)]f₂(x) log y = f₂ (x) log [f₁ (x)] (ii) y = f₁(x) f₂(x) f₃(x) … log y = log [f₁(x) f₂(x) f₃(x) …] log y = log f₁(x) + log Read more about Differentiation Using Logarithm[…]

Differentiation – Some Standard Substitutions Expression Substitution x = asinθ (or) acosθ x = atanθ (or) a cotθ x = asecθ (or) acosecθ (or) x = acosθ (or) acos2θ (or) x = acos²θ + b sin²θ Example: If , then prove that . Solution: Given that, , Let x³ = cosp and y³ = cosq Read more about Differentiation – Some Standard Substitutions[…]

Removable Discontinuity necessarily exists, but is either not equal to f(a) or f(a) is not defined. In this case, it is, possible to redefine the function in such that a manner that and thus, make the function continuous. Consider the function g(x) = (sin x)/x. the function is not defined at x = 0. So, Read more about Removable Discontinuity[…]