Question

Find $d^{2} y / d x^{2}$. $$y=5 x^{3}+4 x$$

   Find $d^{2} y / d x^{2}$.
$$y=5 x^{3}+4 x$$

Calculus and Its Applications

Calculus and Its Applications

Marvin L. Bittinger,… 10th Edition

Chapter 1, Problem 4

Instant Answer

Solved by Expert Gregory Higby

10/22/2020

Step 1

Using the power rule for differentiation, which states that the derivative of $x^n$ is $n*x^{n-1}$, we get: $$\frac{dy}{dx} = 15x^{2}+4$$ Show more…

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Find $d^{2} y / d x^{2}$. $$y=5 x^{3}+4 x$$

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Key Concepts

Power Rule

The power rule is a basic principle in calculus used to differentiate polynomial terms. It states that the derivative of x raised to a power n is n multiplied by x raised to the power of n-1. This rule simplifies the differentiation process for polynomial functions.

Second Derivative

The second derivative is the derivative of the first derivative and is used to assess the concavity of a function. It provides information about the rate of change of the slope, indicating whether the function is concave up or concave down, which is important in understanding the function's curvature and potential inflection points.

First Derivative

The first derivative of a function provides insights into the function's instantaneous rate of change or slope. It is used to determine aspects such as increasing or decreasing behavior and helps lay the groundwork for the computation of higher-order derivatives.

Differentiation

Differentiation is the process in calculus of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input variable. This is a fundamental concept used to analyze the behavior and trends of functions.

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