Differentiate each function. (a) f(x)=√(e^2 x x^3) (b) f(w)=√(e^2 w+w^3)

Name: Problem 24, Chapter 2: Calculus: Early Transcendental Functions
Uploaded: 2022-10-09T02:36:08-08:00
Channel: Sarah Parrigin
Description: Differentiate each function. (a) f(x)=√(e^2 x x^3) (b) f(w)=√(e^2 w+w^3)

step 1 So we're going to find the derivative of f of x is equal to the cube root of e to the 2x x cubed

Differentiate each function. (a) f(x)=√(e^2 x x^3) (b)...

Key Concepts

Rational Exponents and Radical Functions

Radical functions can be rewritten using rational exponents, which simplifies the process of differentiation. For example, a cube root, written as raising to the 1/3 power, allows the application of the power rule in combination with the chain rule. This method streamlines handling functions that involve roots by converting them into an exponentiation form.

Exponential Function Differentiation

Exponential functions, such as those involving e raised to a function of x, have the property that their derivative is proportional to the original function. When the exponent itself is a function of x, the chain rule is applied, and the derivative becomes the exponential function multiplied by the derivative of the exponent. This is a key technique when differentiating functions that include exponential terms.

Chain Rule

The chain rule is used to differentiate composite functions—functions formed by applying one function to the result of another. When a function is written as f(g(x)), its derivative is f'(g(x)) multiplied by g'(x). This is essential when dealing with functions that include an outer function (such as a radical or power) applied to an inner function (which might be a product or a more complex expression).

Product Rule

The product rule is a technique for differentiating the product of two functions. If a function h(x) is given by u(x)v(x), then its derivative h'(x) is u'(x)v(x) + u(x)v'(x). This rule is particularly useful when the inner function in a composite expression is itself a product of two differentiable functions.

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