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Chapter 3: Problem 4

For a given function \(f,\) what does \(f^{\prime}\) represent?

Short Answer

Expert verified
Answer: The symbol \(f^{\prime}\) represents the derivative of a function \(f\) with respect to its variable, indicating the rate of change of the function. For the function \(f(x) = x^2\), the derivative is \(f^{\prime}(x) = 2x\), which tells us the slope of the tangent to the curve at any given point. A positive, negative, or zero derivative signifies increasing, decreasing, or constant function behavior, respectively. In the case of \(f(x) = x^2\), the function is decreasing when \(x < 0\), increasing when \(x > 0\), and reaches a minimum value with a horizontal tangent when \(x = 0\).

Step by step solution

01

Understand the Notation

The symbol \(f^{\prime}\) represents the derivative of the function \(f\). Another common representation for this is \(\frac{df}{dx}\), which means the derivative of the function \(f\) with respect to its variable \(x\). For example, if \(f(x) = x^2\), then its derivative \(f^{\prime}(x)\) would represent how the function \(f(x)\) changes as \(x\) changes.
02

Realise the Significance of The Derivative

The derivative of a function represents its rate of change, or more precisely, the rate at which the function's output changes with respect to changes in its input. In a graphical representation, it tells us the slope of the tangent to the curve at any given point. A positive derivative indicates the function is increasing, while a negative derivative signifies a decreasing function. If the derivative is zero, the function has a horizontal tangent, which could signify a local maximum or minimum.
03

Find the Derivative of a Sample Function

Consider the function \(f(x) = x^2\). Let's find the derivative of this function. We can use the power rule for differentiation: \(\frac{d}{dx}(x^n) = n\cdot x^{n-1}\). Applying the power rule to our function \(f(x) = x^2\), we get: $$ f^{\prime}(x) = 2x^{(2-1)} = 2x $$ Therefore, for the given function \(f(x) = x^2\), its derivative \(f^{\prime}(x) = 2x\). This means that at any given point \(x\) on the curve of this function, the slope of the tangent to the curve is equal to \(2x\).
04

Interpret the Result

The derivative \(f^{\prime}(x) = 2x\) tells us about the rate of change of the function \(f(x) = x^2\). When \(x < 0\), the derivative \(f^{\prime}(x)\) will be negative, indicating a decreasing function. For \(x > 0\), the derivative will be positive, signifying an increasing function. And finally, when \(x = 0\), the derivative \(f^{\prime}(0) = 0\), which implies that at this point, the function has a horizontal tangent and reaches a minimum value.

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Most popular questions from this chapter

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