Find ∂z / ∂u and ∂z / ∂v when u=ln2, v=1 if z=5 tan^-1 x and x=e^n+lnv

Name: Problem 37, Chapter 13: University Calculus: Early Transcendentals
Uploaded: 2023-06-07T05:23:22-08:00
Duration: 2 min 2 s
Channel: Aman Gupta
Description: Find ∂z / ∂u and ∂z / ∂v when u=ln2, v=1 if z=5 tan^-1 x and x=e^n+lnv

step 1 in this problem we have z is equals to 5 tan inverse of x and x is e power of u plus ln of v so

Find ∂z / ∂u and ∂z / ∂v when u=ln2, v=1 if z=5 tan^-1 x and...

Key Concepts

Differentiation of Inverse Trigonometric Functions

Differentiating inverse trigonometric functions requires the use of specific derivative formulas, such as the derivative of the arctan function, which is 1/(1+x^2). This technique is important in problems where the function being differentiated is composed with an inverse trigonometric function, necessitating careful application of the chain rule.

Differentiation of Exponential and Logarithmic Functions

Exponential and logarithmic functions possess well-known differentiation rules—for example, the derivative of an exponential function e^u is e^u, and that of a natural logarithm ln(v) is 1/v. These rules are essential when dealing with composite functions that include exponential and logarithmic terms, as they simplify the process of computing derivatives within the larger structure of the problem.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It states that the derivative of a composed function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In the context of partial differentiation, it allows one to compute how a change in one independent variable indirectly affects a dependent variable through an intermediate function.

Partial Differentiation

Partial differentiation deals with functions of several variables and involves taking the derivative with respect to one variable while keeping the others constant. This concept is crucial when analyzing how changes in one independent variable affect the overall function, especially in cases where the function is defined implicitly through another variable.

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