Find (d y)/(d x), if y=(tanx)^cotx+(cotx)^lan x

Name: Problem 72, Chapter 5: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-03-07T12:32:23-08:00
Duration: 1 min 48 s
Channel: Rukhmani Jain
Description: Find (d y)/(d x), if y=(tanx)^cotx+(cotx)^lan x

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a method used to differentiate functions where both the base and the exponent depend on the variable. By taking the natural logarithm of both sides of the equation, the exponent can be brought down as a multiplier, converting a complex power function into a simpler form that can be differentiated using standard rules.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of an inner function u(x) and an outer function f(u), then the derivative of y with respect to x is given by the derivative of f with respect to u times the derivative of u with respect to x.

Product Rule

The product rule is used to differentiate the product of two functions. If y is the product of two functions, say u(x) and v(x), then the derivative of y is found by differentiating u and v separately and combining them as u'(x)v(x) + u(x)v'(x). This rule is especially useful when logarithmic differentiation leads to the differentiation of products.

Differentiation of Exponential Functions with Variable Exponents

Differentiating exponential functions where both the base and the exponent are functions of the variable requires a specialized approach. In such cases, logarithmic differentiation transforms the expression into a form where the exponent multiplies the logarithm of the base, after which the standard rules of differentiation (including the chain and product rules) can be applied effectively.

Differentiation of Trigonometric Functions

Differentiation of trigonometric functions covers finding the rates of change of functions such as sine, cosine, tangent, and their reciprocals. Familiarity with the derivatives of these functions is essential, as they frequently appear in problems involving variable exponentials and composite functions.

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