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Magazine Chapter 23: Problem 15
In Exercises \(3-26,\) find the derivative of each of the functions by using the definition. $$y=5 x^{3}+4 x-3 \pi^{2}$$
Short Answer
Expert verified
The derivative of \( y = 5x^3 + 4x - 3\pi^2 \) is \( 15x^2 + 4 \).
Step by step solution
01
Understand the Definition of Derivative
The derivative of a function at a point is the limit of the difference quotient as the interval approaches zero. Mathematically, this is given by the formula: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\] We will use this definition to find the derivative of the function.
02
Substitute the Function into the Definition
Given the function \( y = 5x^3 + 4x - 3\pi^2 \), express \( f(x+h) \) as follows:\[ f(x+h) = 5(x+h)^3 + 4(x+h) - 3\pi^2 \]Now expand and simplify this expression.
03
Expand \( (x+h)^3 \)
Expanding \( (x+h)^3 \) using the binomial theorem:\[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \]Substitute back into the function, giving:\[ f(x+h) = 5 \left(x^3 + 3x^2h + 3xh^2 + h^3\right) + 4(x+h) - 3\pi^2 \] Simplify the expression.
04
Simplify \( f(x+h) \)
Substitute the expanded \( (x+h)^3 \) into the function and simplify:\[ f(x+h) = 5x^3 + 15x^2h + 15xh^2 + 5h^3 + 4x + 4h - 3\pi^2 \]Now find the difference \( f(x+h) - f(x) \).
05
Find \( f(x+h) - f(x) \) and Simplify
Subtract \( f(x) = 5x^3 + 4x - 3\pi^2 \) from \( f(x+h) \):\[ f(x+h) - f(x) = (5x^3 + 15x^2h + 15xh^2 + 5h^3 + 4x + 4h - 3\pi^2) - (5x^3 + 4x - 3\pi^2) \]Cancel out common terms:\[ = 15x^2h + 15xh^2 + 5h^3 + 4h \] This expression can now be used to form the difference quotient.
06
Form and Simplify the Difference Quotient
Divide the expression by \( h \) to form the difference quotient:\[ \frac{15x^2h + 15xh^2 + 5h^3 + 4h}{h} = 15x^2 + 15xh + 5h^2 + 4 \]Now take the limit as \( h \to 0 \).
07
Apply the Limit as \( h \to 0 \)
Calculate the limit of the difference quotient as \( h \to 0 \):\[ \lim_{h \to 0} (15x^2 + 15xh + 5h^2 + 4) = 15x^2 + 4 \]As \( h \to 0 \), the terms involving \( h \) vanish.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
difference quotient
The difference quotient is a fundamental aspect of understanding derivatives. It acts as a bridge between the function and its derivative, helping us understand rates of change. Imagine it as a formula capturing the average rate of change of a function over a small interval. The mathematical expression is:
- \[ \frac{f(x+h) - f(x)}{h} \]
binomial theorem
The binomial theorem is a valuable tool in the process of expanding expressions raised to a power, especially in calculus exercises. It's essential when you encounter terms like
- \((x+h)^3\)
- \[(a + b)^n = \sum_{k=0}^{n} \begin{pmatrix} n \ k \end{pmatrix} a^{n-k}b^k\]
limit calculation
Limit calculation is the cornerstone of derivative definition. It's the process that allows us to equate the difference quotient to the derivative. A limit describes the behavior of a function as it nears a certain point or approaches infinity. In calculating limits, we express:
- \[ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
calculus exercises
Calculus exercises are essential for mastering differentiation and integration, the primary operations in calculus. They range from simple polynomial differentiation to complex multi-variable functions. In our exercise, we used basic calculus tools such as the difference quotient, binomial expansion, and limit calculations to find the derivative of a polynomial function.
These exercises often involve:
These exercises often involve:
- Understanding function behavior
- Applying mathematical theorems like the binomial theorem
- Precision in calculation
