Find the derivatives of the following functions. f(x)=x^2 cosh^2 3 x

Name: Problem 29, Chapter 7: Calculus: Early Transcendentals
Uploaded: 2020-05-25T23:25:55-08:00
Duration: 58 s
Channel: Amy Jiang
Description: Find the derivatives of the following functions. f(x)=x^2 cosh^2 3 x

step 1 Okay, so let's take the derivative of the following. Notice here that we have x squared as well

Find the derivatives of the following functions. f(x)=x^2...

Key Concepts

Hyperbolic Functions Differentiation

Hyperbolic functions, such as cosh and sinh, have properties similar to trigonometric functions but are defined using exponential functions. Their derivatives follow specific rules, for example, the derivative of cosh(x) is sinh(x), which is essential in differentiating expressions involving hyperbolic functions, especially when combined with other differentiation rules.

Chain Rule

The Chain Rule is a powerful method for differentiating composite functions, where one function is applied inside another. It dictates that the derivative of a composite function f(g(x)) is the derivative of f with respect to g, evaluated at g(x), multiplied by the derivative of g with respect to x, allowing for the systematic differentiation of nested functions.

Power Rule

The Power Rule is a fundamental differentiation technique used when the function is a power of the variable. For any function of the form x^n, the derivative is given by multiplying by the exponent and reducing the exponent by one, a method that simplifies the process of differentiating polynomial expressions.

Differentiation

Differentiation is the process of computing the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. It forms the basis of differential calculus and is essential for understanding and modeling change in various applied fields.

Product Rule

The Product Rule is used to differentiate functions that are the product of two or more functions. It states that the derivative of a product f(x) * g(x) is f'(x) * g(x) plus f(x) * g'(x). This rule is critical when handling functions where individual factors are themselves functions of the variable.

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