Unit 3: Differentiability of function - Subjective Questions

Derivative by First Principle:

The derivative of a function with respect to is defined as:

provided the limit exists.

Finding derivative of :

Substituting:

Therefore, .

Using first principle:

Using the identity :

Rewriting:

Since and :

Therefore, .

Let and be two differentiable functions. Then:

1. Sum Rule:

2. Difference Rule:

3. Product Rule:

4. Quotient Rule:

5. Constant Multiple Rule:

These rules allow us to differentiate complex algebraic combinations of functions systematically.

Differentiability at a point:

A function is said to be differentiable at if the following limit exists:

This means the left-hand derivative must equal the right-hand derivative:

Relationship between Continuity and Differentiability:

• If a function is differentiable at a point, then it is continuous at that point.
• However, the converse is not always true. A function can be continuous but not differentiable.
• Example: is continuous at but not differentiable there (the graph has a sharp corner).

Conclusion: Differentiability Continuity, but Continuity Differentiability.

Checking Continuity at :

Since the left limit = right limit = , the function is continuous at .

Checking Differentiability at :

Left-hand derivative (LHD):

Right-hand derivative (RHD):

Since LHD RHD , the derivative does not exist at .

Conclusion: is continuous but not differentiable at .

Chain Rule:

If and are both differentiable functions, then the derivative of the composite function is:

Applying to :

Let , so .

Therefore:

Derivatives of Composite Functions:

A composite function is a function of a function, written as . To differentiate it, we use the chain rule, differentiating the outer function while keeping the inner function intact, then multiplying by the derivative of the inner function.

Solving :

Let , so .

Therefore:

Implicit Differentiation:

When a relationship between and is given in the form and cannot be easily solved for explicitly, we differentiate both sides with respect to , treating as a function of . Wherever we differentiate a term in , we multiply by (chain rule).

Solving :

Differentiate both sides with respect to :

Solving for :

Derivative of :

Let , so , where .

Differentiating both sides with respect to :

Since (positive in the given range):

Derivative of :

Since , differentiating both sides:

Derivative of :

Let , so .

Differentiating:

Therefore:

Derivative of :

Let , so .

Differentiating:

Since and :

Derivative of from first principles:

Using the standard limit :

Therefore, .

Derivative of :

Using :

Derivative of :

Let , then .

Differentiating both sides with respect to :

Therefore:

Derivative of :

Using the change of base formula :

Logarithmic Differentiation:

This technique is used to differentiate functions of the form (variable base and variable exponent) or products/quotients of many functions. We take the natural logarithm on both sides to simplify the expression before differentiating.

Steps:

• Take on both sides.
• Use log properties to simplify.
• Differentiate implicitly.
• Solve for .

Solving :

Taking on both sides:

Differentiating both sides with respect to :

Therefore:

Given:

Taking natural logarithm on both sides:

Differentiating both sides with respect to (using product rule on the right):

Therefore:

Parametric Differentiation:

When both and are expressed in terms of a third variable (parameter), say or , i.e., and , then the derivative is found using:

Solving , :

Differentiating with respect to :

Therefore:

Given: ,

First derivatives with respect to :

First derivative :

Second order derivative :

Therefore, .

Second Order Derivative:

The second order derivative of a function is the derivative of its first derivative. If , then:

• First derivative:
• Second derivative:

It represents the rate of change of the slope and is useful in determining concavity and points of inflection.

Solving :

First derivative:

Second derivative:

Explicit Function:

• A function where the dependent variable is expressed directly and solely in terms of the independent variable .
• Form:
• Example: ,

Implicit Function:

• A function where and are mixed together in an equation and is not isolated.
• Form:
• Example: ,

Comparison Table:

When Implicit Differentiation is Preferred:

Implicit differentiation is preferred when it is difficult or impossible to solve for explicitly (e.g., ), or when isolating makes differentiation more complex. It allows finding without solving for .

Given: (product of three functions)

Using the extended product rule for three functions :

Let:

• (chain rule)

Substituting:

Factoring out :

Given:

First derivative (product rule):

Note that , so .

Second derivative (differentiate equation (1)):

Since , we have and :

Rearranging:

Hence proved.