Differentiate. g(x)=(5 x^2-8 x) e^x^2-4 x

Name: Problem 54, Chapter 3: Calculus and its applications
Uploaded: 2020-11-02T00:01:01-08:00
Duration: 2 min 49 s
Channel: Jonathan Mezzenga
Description: Differentiate. g(x)=(5 x^2-8 x) e^x^2-4 x

Key Concepts

Product Rule

The product rule is used when differentiating functions that are the product of two or more differentiable functions. It states that the derivative of the product is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. This rule allows each part of the product to be handled individually and then combined to form the complete derivative.

Chain Rule

The chain rule is essential for differentiating composite functions, where one function is nested within another. It indicates 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 particularly important when the exponent of an exponential function is itself a function of x.

Exponential Function Differentiation

Differentiating an exponential function that has a function in its exponent requires using the chain rule. The derivative of an exponential function of the form e^(u(x)) is given by e^(u(x)) multiplied by u'(x), where u(x) is a differentiable function. This concept is crucial when the exponent is not a simple variable but a more complex expression.

Polynomial Differentiation

Differentiating polynomial functions involves applying the power rule, which states that the derivative of x raised to any power n is n times x raised to the power of (n-1). Each term in a polynomial is differentiated separately, and the results are summed up to obtain the overall derivative. This rule provides a straightforward method to differentiate any polynomial expression.

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