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Chapter 3: Problem 12

Find \(D_{x} y\) using the rules of this section. $$ y=3 x^{4}+x^{3} $$

Short Answer

Expert verified
The derivative \( D_x y = 12x^3 + 3x^2 \).

Step by step solution

01

Identify Terms

The function given is \( y = 3x^4 + x^3 \). This equation consists of two polynomial terms: \( 3x^4 \) and \( x^3 \).
02

Apply Power Rule to Each Term

Apply the power rule for differentiation which states \( \frac{d}{dx}[x^n] = nx^{n-1} \). For \( 3x^4 \), it becomes \( 3 \times 4x^{4-1} = 12x^3 \). For \( x^3 \), it becomes \( 3x^{3-1} = 3x^2 \).
03

Combine the Derivatives

Add the derivatives of the two terms together. Thus, \( D_x y = 12x^3 + 3x^2 \).

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Key Concepts

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

Understanding the Power Rule
One of the most fundamental rules in calculus for differentiation is the power rule. It applies to functions that are polynomials or contain terms with variables raised to an exponent. The power rule simplifies how we find the derivative, which measures how a function changes as its input changes.

The power rule states:
  • If you have a term in the form of \(x^n\), the derivative is \(nx^{n-1}\).
  • This means you multiply the power by the coefficient and then decrease the power by one.
Using the power rule makes it easier to handle complex polynomials by breaking them down into their individual terms, differentiating each, and then recombining.
Breaking Down Polynomials
Polynomials are expressions consisting of variables and coefficients, structured with operations of addition, subtraction, multiplication, and non-negative integer exponents. In simpler terms, they're composed of several terms added or subtracted together, each term featuring a variable raised to some power.

For example, the polynomial given in the exercise is \(y = 3x^4 + x^3\). It has two terms:
  • \(3x^4\), where 3 is the coefficient and 4 is the exponent.
  • \(x^3\), where the coefficient is 1 (implied) and 3 is the exponent.
Understanding the structure of polynomials helps when applying differentiation rules like the power rule because each term can be evaluated separately.
Calculating Derivatives
Derivatives are a central part of calculus, representing the rate at which a function changes at any given point. Finding a derivative involves applying differentiation rules, like the power rule, to each term of the polynomial.
  • For the term \(3x^4\), use the power rule to get \(12x^3\).
  • For \(x^3\), applying the same rule gives \(3x^2\).
These steps help you find the derivative of a polynomial expression, providing a new function that tells you the slope or rate of change at any point on the original graph.
The Broader Context of Calculus
Calculus is a branch of mathematics that studies how things change. It involves concepts like derivatives to analyze how quantities vary and integrate different mathematical insights to develop deep understanding.
  • In the context of our exercise, using differentiation allows us to understand how the expression \(y = 3x^4 + x^3\) changes as \(x\) changes.
  • By finding the derivative \(D_x y = 12x^3 + 3x^2\), we gain insight into the behavior of the function, such as identifying increasing or decreasing trends.
Calculus connects such mathematical techniques to real-world scenarios, like calculating speed, optimizing functions, and understanding dynamic systems.

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Most popular questions from this chapter

Suppose \(f\) is differentiable. If we use the approximation \(f(x+h) \approx f(x)+f^{\prime}(x) h\) the error is \(\varepsilon(h)=f(x+h)-\) \(f(x)-f^{\prime}(x) h .\) Show that(a) \(\lim _{h \rightarrow 0} \varepsilon(h)=0\) and (b) \(\lim _{h \rightarrow 0} \frac{\varepsilon(h)}{h}=0\).

Suppose that curves \(C_{1}\) and \(C_{2}\) intersect at \(\left(x_{0}, y_{0}\right)\) with slopes \(m_{1}\) and \(m_{2}\), respectively, as in Figure 4 . Then (see Problem 40 of Section \(1.8\) ) the positive angle \(\theta\) from \(C_{1}\) (i.e., from the tangent line to \(C_{1}\) at \(\left.\left(x_{0}, y_{0}\right)\right)\) to \(C_{2}\) satisfies $$ \tan \theta=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$ Find the angles from the circle \(x^{2}+y^{2}=1\) to the circle \((x-1)^{2}+y^{2}=1\) at the two points of intersection.

Suppose that \(f\) is differentiable and that there are real numbers \(x_{1}\) and \(x_{2}\) such that \(f\left(x_{1}\right)=x_{2}\) and \(f\left(x_{2}\right)=x_{1}\). Let \(g(x)=f(f(f(f(x))))\). Show that \(g^{\prime}\left(x_{1}\right)=g^{\prime}\left(x_{2}\right)\).

Find all points on the curve \(x^{2} y-x y^{2}=2\) where the tangent line is vertical, that is, where \(d x / d y=0\).

Where does the tangent line to \(y=\left(x^{2}+1\right)^{-2}\) at \(\left(1, \frac{1}{4}\right)\) cross the \(x\) -axis?
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