Differentiate. g(x)=(4 x^2+3 x) e^x^2-7 x

Name: Problem 53, Chapter 3: Calculus and its applications
Uploaded: 2020-11-01T23:56:48-08:00
Duration: 3 min 25 s
Channel: Jonathan Mezzenga
Description: Differentiate. g(x)=(4 x^2+3 x) e^x^2-7 x

Key Concepts

Derivative of Exponential Function

Differentiating an exponential function, especially those of the form e^(f(x)), requires the application of the chain rule. The derivative of e^(f(x)) is equal to e^(f(x)) multiplied by the derivative of f(x). This rule simplifies handling exponential functions with complex exponents in the differentiation process.

Polynomial Differentiation

Polynomial differentiation involves finding the derivative of polynomial expressions by applying the power rule, which states that the derivative of x^n is n*x^(n-1). This process is straightforward and is used to differentiate polynomial parts of more complex functions, serving as the basis for applying further rules such as the product and chain rules.

Product Rule

The product rule is a fundamental differentiation technique used when differentiating two functions multiplied together. It states that the derivative of the product of two functions is given by the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function. This is essential when handling expressions that are products, such as a polynomial times an exponential function.

Chain Rule

The chain rule is used to differentiate composite functions where one function is nested within another. When a function is applied to the result of another function, the derivative is found by taking the derivative of the outer function evaluated at the inner function and multiplying it by the derivative of the inner function. This is particularly important for differentiating the exponent in an exponential function when its exponent is itself a function of x.

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