Question

Calculate the derivative. $$ f(x)=3 e^{x}-x^{3} $$

   Calculate the derivative.
$$
f(x)=3 e^{x}-x^{3}
$$

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 3, Problem 30

Instant Answer

Solved by Expert Elyse Gonzalez

05/20/2019

Step 1

The first term is $3 e^{x}$ and the second term is $-x^{3}$. Show more…

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Calculate the derivative. $$ f(x)=3 e^{x}-x^{3} $$

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Key Concepts

Differentiation

Differentiation is the process of finding the rate at which a function changes at any point. It involves computing the derivative, which represents the slope of the tangent line to the function's graph, and is essential for analyzing the behavior of functions in various contexts such as optimization and motion problems.

Exponential Function

An exponential function is one in which the variable appears in the exponent, typically represented as e^x in calculus. Its derivative is unique in that it is the same as the original function, making it a fundamental component in solving differential equations and modeling growth or decay processes.

Power Rule

The power rule is a basic principle used in differentiation that states if a function is of the form x^n, its derivative is nx^(n-1). This rule simplifies the process of differentiating polynomial expressions and is widely used in calculus for functions involving variables raised to a power.

Linearity of Differentiation

Linearity of differentiation refers to the property that the derivative of a sum of functions is the sum of the derivatives of each individual function, and similarly, the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This principle allows for the straightforward differentiation of complex expressions by applying it term by term.

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