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Chapter 12: Problem 33

Find the derivative of the function. \(f(x)=4-3 x^{2}\)

Short Answer

Expert verified
The derivative of the function \(f(x) = 4 - 3x^{2}\) is \(-6x\).

Step by step solution

01

Identify the Power Rule

The power rule is a basic theorem in calculus used to find the derivative of a function that is a power of x. The power rule is \(d/dx [x^n] = n*x^{(n-1)}\). In our case, we will apply this rule to the term \(-3x^{2}\), where \(n=2\). The constant term does not affect the derivative, as the derivative of a constant is always zero.
02

Apply the Power Rule

Applying the power rule to \(-3x^{2}\) gives us \(-3*2*x^{(2-1)}\). Simplifying this expression yields \(-6x\). The derivative of the constant term, 4, is zero.
03

Combine the Results

The final derivative of the function is the sum of the derivative of the individual terms. As found in step 2, the derivative of \(-3x^{2}\) is \(-6x\) and the derivative of 4 is 0. So, summing these up gives us \(-6x + 0\), or just \(-6x\).

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Key Concepts

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

Derivative
A derivative reflects how a function changes as its input changes. It's like a mathematical speedometer, indicating how fast or slow a quantity is changing. In this exercise, we focused on using the power rule, which simplifies the process of finding derivatives of polynomial functions.

To derive a term involving a power of x, such as \\(-3x^2\), we apply the power rule: \\( \frac{d}{dx} [x^n] = n \cdot x^{n-1} \). This rule tells us to multiply the exponent by the coefficient, then decrease the exponent by one.

For the function \\( f(x) = 4 - 3x^2 \), we used this rule to find the derivative of \\(-3x^2\), resulting in \\(-6x\). Remember that each term's derivative is calculated independently, then combined to find the overall derivative.
Constant Term
Constant terms are numbers without variables attached. They stay the same regardless of the value of x. In calculus, the derivative of a constant term, like 4 in our function, is always zero.

This happens because a constant does not change; its graph is a horizontal line, and so its rate of change, or slope, is zero. When calculating derivatives, such as for \\( f(x) = 4 - 3x^2 \), constants simply vanish from the derivative. This means that in our step-by-step solution, the term 4 contributes nothing to the site's overall derivative, simplifying our calculations.
Simplifying Expressions
Simplifying expressions is an essential final step in deriving functions. Once the derivatives of each term are found, they should be combined and simplified to present the solution in the most understandable form.

In the exercise, after applying the power rule and finding the derivative \\(-6x\), as well as noting that the constant term 4 gives zero, we combine these results for the final derivative.
  • The derivative of \\(-3x^2\) is \\(-6x\).
  • The derivative of the constant 4 is zero.
  • Thus, we combine these to find \\(-6x + 0\), simplifying to just \\(-6x\).
This simplification step ensures that the derivative is clear and concise, emphasizing the change in the function due to the variable term only.

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Most popular questions from this chapter

Numerical and Graphical Analysis In Exercises \(35 - 38 ,\) (a) complete the table and numerically estimate the limit as \(x\) approaches infinity, and (b) use a graphing utility to graph the function and estimate the limit graphically. $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & { 10 ^ { 0 } } & { 10 ^ { 1 } } & { 10 ^ { 2 } } & { 10 ^ { 3 } } & { 10 ^ { 4 } } & { 10 ^ { 5 } } & { 10 ^ { 6 } } \\ \hline f ( x ) & { } & { } & { } & { } & { } & { } \\ \hline \end{array}$$ $$ f ( x ) = 3 x - \sqrt { 9 x ^ { 2 } + 1 } $$

Evaluating a Limit at Infinity In Exercises \(9 - 28\) , find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify your result graphically. $$ \lim _ { y \rightarrow \infty } \frac { 4 y ^ { 4 } } { y ^ { 2 } + 3 } $$

Finding the Limit of a Sequence In Exercises \(55 - 58\) , find the limit of the sequence. Then verify the limit numerically by using a graphing utility to complete the table. $$\begin{array} { | c | c | c | c | c | c | c | } \hline n & { 10 ^ { 0 } } & { 10 ^ { 1 } } & { 10 ^ { 2 } } & { 10 ^ { 3 } } & { 10 ^ { 4 } } & { 10 ^ { 5 } } & { 10 ^ { 6 } } \\ \hline a _ { n } & { } & { } & { } & { } & { } & { } \\\ \hline \end{array}$$ $$ a _ { n } = \frac { 4 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 2 } \right] \right) $$

Use the function and its derivative to determine any points on the graph of \(f\) at which the tangent line is horizontal. Use a graphing utility to verify your results. \(f(x)=x^{4}-2 x^{2}, \quad f^{\prime}(x)=4 x^{3}-4 x\)

Think About It (a) When \(f(2)=4,\) can you conclude anything about \(\lim _{x \rightarrow 2} f(x) ?\) Explain your reasoning. (b) When \(\lim _{x \rightarrow 2} f(x)=4,\) can you conclude anything about \(f(2) ?\) Explain your reasoning.
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