Maths Continuity and Differentiability

Some of the common mistakes that people usually make while using logarithmic differentiation have been mentioned below:

• Not Multiplying by y: After logarithmic differentiation, it is mandatory to multiply by y to solve for dy/dx?
• Incorrectly Applying the Chain Rule: Make sure that you have correctly used the chain rule whenever you are differentiating a logarithmic expression.
• Using Wrong Logarithm: It is always advisable to use the natural logarithm (ln) instead of logarithms with other bases.
• Ignoring Domain Restrictions: Natural logarithm is only defined for the positive real numbers; therefore, y>0 whenever you apply logarithmic differentiation.

Logarithmic differentiation is used in the following cases:

• Logarithmic differentiation is used with functions that have a variable in both base and exponents. In such a case, standard differentiation rules do not apply directly to such functions. This differentiation converts exponentiation into multiplication.
• Another area where logarithmic differentiation is used is with a function which is the product of a quotient of multiple terms. 
• Whenever a function has a complex combination of multiplication, division and exponentiation, logarithmic differentiation is preferred. This differentiation eases the complexity by converting multiplicative relations into additive ones that are easier to handle.

We use parametric equations for the following reasons:

• Parametric equations represent all those curves that are otherwise impossible to be represented as a single function. Circles, cycloids, ellipses and spirals are all described using parametric equations.
• These equations can easily describe motion of objects over time.
• Parametric equations help in breaking down complex relationships into simpler components. Rather than dealing with single complex equation, one can describe x and y seperately in terms of t parameter.
• These equations extend to three dimensions easily and naturally, so that one can describe curves and surfaces in 3D space.