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Chapter 2: Problem 5

Find the derivative of the function. \(y=x^{6}\)

Short Answer

Expert verified
The derivative of the function \(y = x^{6}\) is \( \frac{dy}{dx} = 6x^{5}\).

Step by step solution

01

Identify the function

First, identify the function that needs to be differentiated. In this exercise the function is \(y = x^{6}\).
02

Apply the power rule

Next, apply the power rule of differentiation. The power rule states that the derivative of \(x^{n}\) is \(n \cdot x^{n-1}\). If we apply it to our function, we get \( \frac{dy}{dx} = 6 \cdot x^{6-1}\).
03

Simplify the derivative

Now, simplify the resulting expression for the derivative of the function. Simplifying gives \( \frac{dy}{dx} = 6 \cdot x^{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The Power Rule is a fundamental principle in differentiation and is especially useful when working with polynomial functions. Its main idea is to simplify the process of finding derivatives for functions where the variable is raised to a power.
  • The Power Rule states that if you have a function of the form \(y = x^n\), then the derivative is given by multiplying the exponent \(n\) by \(x^{n-1}\).
  • This means you bring down the power as a coefficient and then subtract one from the exponent.
For example, to differentiate \(y = x^6\), apply the Power Rule and you get \(dy/dx = 6x^{5}\). This simplification is handy when working with any polynomial function and is one of the first differentiation rules taught because of its straightforwardness and utility.
Derivative
The derivative of a function is a key concept in calculus and represents the rate at which a function is changing at any given point. It provides a way to calculate how a quantity changes as another quantity changes.
  • The derivative tells you the slope of the tangent line to the graph of the function at any point.
  • Using derivatives, you can predict trends and make approximations about the function's behavior.
In the context of polynomial functions, derivatives help us understand how these functions grow or shrink as their variables change. For instance, with \(y = x^6\), the derivative \(dy/dx = 6x^5\) shows us that the rate of change is not constant but depends on the value of \(x\). The larger the \(x\), the greater the rate of increase of the function.
Polynomial Function
Polynomial functions are algebraic expressions consisting of variables raised to powers and coefficients. They can range from simple to complex, but understanding them is critical in calculus and other advanced mathematics.
  • These functions are expressed as a sum of terms of the form \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\).
  • Each term in a polynomial function is made up of a coefficient \(a\) and a variable \(x\) raised to a non-negative integer power.
  • Polynomials are distinguished by their degree, which is the highest power of \(x\) present in the expression.
For example, \(y = x^6\) is a simple polynomial function, and its highest power, which determines its degree, is 6. Understanding polynomial functions is important for finding derivatives, as applying rules like the Power Rule becomes essential in handling them efficiently.

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Most popular questions from this chapter

Think About It Describe the relationship between the rate of change of \(y\) and the rate of change of \(x\) in each expression. Assume all variables and derivatives are positive. (a) \(\frac{d y}{d t}=3 \frac{d x}{d t}\) (b) \(\frac{d y}{d t}=x(L-x) \frac{d x}{d t}, \quad 0 \leq x \leq L\)

Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpendicular to each other.] $$ \begin{aligned} &x+y=0 \\ &x=\sin y \end{aligned} $$

Find the points at which the graph of the equation has a vertical or horizontal tangent line. $$ 25 x^{2}+16 y^{2}+200 x-160 y+400=0 $$

Find the second derivative of the function. $$ f(x)=\frac{1}{x-2} $$

When a certain polyatomic gas undergoes adiabatic expansion, its pressure \(p\) and volume \(V\) satisfy the equation \(p V^{1.3}=k\), where \(k\) is a constant. Find the relationship between the related rates \(d p / d t\) and \(d V / d t\).
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