Question

Differentiate. $$ g(x)=-\frac{4}{5} e^{x^{3}} $$

   Differentiate.
$$
g(x)=-\frac{4}{5} e^{x^{3}}
$$

Calculus and its applications

Calculus and its applications

Bittinger, Marvin L.… 11th Edition

Chapter 3, Problem 26

Instant Answer

Solved by Expert Jonathan Mezzenga

10/31/2020

Step 1

In this case, we have g(x) = -4/5 * e^(x^3). We can think of this as f(u) = -4/5 * e^u, where u = x^3. Then, we need to find the derivatives of f(u) and u with respect to x. The derivative of f(u) with respect to u is: f'(u) = -4/5 * e^u The derivative of u = Show more…

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Differentiate. $$ g(x)=-\frac{4}{5} e^{x^{3}} $$

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Key Concepts

Chain Rule

The chain rule is an essential method for differentiating composite functions, where one function is nested inside another. It states that to differentiate a composite function, you first differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes as its input changes. It is a key concept in calculus used to analyze the behavior of functions, find slopes of tangents, and solve real-world problems involving change.

Exponential Function Differentiation

Exponential function differentiation involves finding the derivative of functions where the variable appears in the exponent. These functions typically have the form e^(u(x)), and the differentiation process often requires evaluating the derivative of the exponent as part of the procedure.

Constant Multiple Rule

The constant multiple rule is a basic principle of differentiation which asserts that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. This rule simplifies the process of differentiating functions that are scaled by a constant factor.

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