Find the derivatives of (a) y=tanx^x (b) y=x^x e^x^2 (c) y=x^x^x.

Name: Problem 61, Chapter 3: Calculus, Early Transcendentals
Uploaded: 2021-08-13T21:40:43-08:00
Duration: 1 min 4 s
Channel: Carson Merrill
Description: Find the derivatives of (a) y=tanx^x (b) y=x^x e^x^2 (c) y=x^x^x.

step 1 For the given problem, we want to take the derivative of f of x, which is equal to the tangent o

Find the derivatives of (a) y=tanx^x (b) y=x^x e^x^2 (c)...

Key Concepts

Differentiation of Exponential Functions

This involves rules and techniques for differentiating functions where the variable appears in the exponent. It combines the chain rule with the differentiation of the exponential function, especially when the exponent itself is a complex function of the variable. This is crucial when handling expressions like e^(x^2) or dealing with forms such as x^x.

Product Rule

A rule used for differentiating products of two or more functions. It states that the derivative of the product of two functions is given by the derivative of the first function times the second plus the first function times the derivative of the second. This concept is particularly relevant when the function to be differentiated is expressed as a product of separate functions.

Chain Rule

A fundamental rule in calculus for differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This rule is essential when functions are nested within each other, as is common in expressions involving exponentials and trigonometric functions.

Logarithmic Differentiation

A technique used to differentiate functions where both the base and the exponent depend on the variable, such as functions of the form f(x)^(g(x)). By taking the natural logarithm of both sides, the exponent can be brought down as a multiplier, which simplifies the differentiation process through implicit differentiation.

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