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Chapter 1: Problem 6

\(f(x)=20 x^{2}\) at \(x=a\)

Short Answer

Expert verified
The derivative of the function \(f(x)=20 x^{2}\) at \(x=a\) is \(f'(a) = 40*a\).

Step by step solution

01

Applying the Power Rule

Apply the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{(n-1)}\). So, differentiate \(f(x)=20 x^{2}\) and we get \(f'(x) = 2*20 * x^{(2-1)}\), which simplifies to \(f'(x) = 40x\).
02

Substitute x with a

Substitute \(x\) with \(a\) in the differentiated function to find the derivative at \(x=a\). So, \(f'(a) = 40*a\).

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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 technique in calculus for differentiating functions. This rule provides a quick and straightforward way to find the derivative of a polynomial function. It states that if you have a function of the form \(x^n\), then its derivative is \(nx^{n-1}\). Here鈥檚 a simple breakdown:
  • The derivative of a constant times a power of \(x\) is calculated by multiplying the exponent by the coefficient and then decreasing the exponent by one.
  • In our example, with the function \(f(x)=20x^2\), applying the power rule results in multiplying the exponent \(2\) by the coefficient \(20\).
  • Thus, we get \(2 \cdot 20 = 40\) and reduce the exponent \(2\) by one to get \(x^1\) or simply \(x\).
Applying this rule helps in calculating derivatives efficiently, making it easier to handle more complex functions in calculus.
Derivative
A derivative in calculus signifies the rate at which a function is changing at any given point. Understanding this concept is crucial as it helps in grasping how functions behave.The derivative expresses how the output of a function changes as its input changes. Let's take our function \(f(x)=20x^2\) again:
  • The calculated derivative, \(f'(x) = 40x\), indicates the slope of the tangent to the curve at any point \(x\).
  • It tells us how steep the graph of the function is at every point \(x\).
  • This is a powerful tool as it can help predict future behavior and find points where the function reaches maxima or minima, among other applications.
Understanding derivatives allows you to solve various problems in physics, engineering, and economics by analyzing how variables change with respect to one another.
Function substitution
Function substitution is an important step in evaluating derivatives or functions at specific points. This concept involves replacing the variable in a function with a given number or another expression.Consider the function \(f'(x) = 40x\) from our example:
  • We need to find the derivative of \(f\) at a specific point \(x=a\).
  • To do this, substitute \(x\) with \(a\), resulting in \(f'(a) = 40a\).
  • This substitution helps determine the rate of change at \(x=a\) specifically, rather than at any arbitrary point.
Function substitution is not only used in differentiation but also in integration and solving equations. It鈥檚 a versatile tool for calculating and simplifying mathematical expressions.

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