Problem 92 - Calculus for Biology and Medicine

Question

Assume that \(f(x)\) is a differentiable function. Find the derivative of the reciprocal function \(g(x)=1 / f(x)\) at those points \(x\) where \(f(x) \neq 0\)

Step-by-Step Solution

Answer

The derivative of \(g(x) = \frac{1}{f(x)}\) is \(g'(x) = \frac{-f'(x)}{(f(x))^2}\).

Step 1: Understanding the Reciprocal Function

We begin by recognizing the function given as the reciprocal function for which we need to find the derivative. We have \(g(x) = \frac{1}{f(x)}\), where \(f(x)\) is a differentiable function and \(f(x) eq 0\).

Step 2: Using the Quotient Rule

To find \(g'(x)\), the derivative of \(g(x)\), we can use the quotient rule. The quotient rule states that if \(h(x) = \frac{u(x)}{v(x)}\), then \(h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). For \(g(x)\), \(u(x) = 1\) and \(v(x) = f(x)\).

Step 3: Applying the Quotient Rule Terms

With \(u(x) = 1\) and \(v(x) = f(x)\), we find their derivatives: \(u'(x) = 0\) (since the derivative of a constant is zero) and \(v'(x) = f'(x)\). Substitute these into the quotient rule formula: \[g'(x) = \frac{(0)f(x) - (1)f'(x)}{(f(x))^2} = \frac{-f'(x)}{(f(x))^2}\].

Step 4: Interpreting the Derivative

The derivative \(g'(x) = \frac{-f'(x)}{(f(x))^2}\) shows how the reciprocal function \(g(x)\) changes with respect to \(x\). It is defined only where \(f(x) eq 0\).

Key Concepts

Reciprocal Function

A reciprocal function is simply the inverse of another function. When you have a function such as \( f(x) \), its reciprocal function is given by \( g(x) = \frac{1}{f(x)} \). In simpler terms, if you think about dividing 1 by the function's value, you can visualize the reciprocal. This means if \( f(x) \) outputs a large number, \( g(x) \) will be small and vice versa.

• The reciprocal function \( g(x) \) flips the behavior of the original function around. For instance, where \( f(x) \) increases, \( g(x) \) may decrease.
• It's important to remember that a reciprocal function is only defined where the original function \( f(x) \) is not equal to zero, because division by zero is undefined.

This characteristic makes reciprocal functions interesting but also kind of tricky to handle in calculus, especially when it comes to differentiation.

Derivative

The derivative is a core concept in calculus that measures how a function changes as its input changes. Mathematically, it provides the slope of the tangent line to the function at any given point. This means it tells us how steep the function is at any point. In the context of the exercise, we are finding the derivative of the reciprocal function, \( g(x) = \frac{1}{f(x)} \).Using the quotient rule, which comes in handy when dividing one function by another, allows you to find the derivative. The quotient rule formula is:\[ h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\]For our reciprocal function:- \( u(x) = 1 \) with \( u'(x) = 0 \)- \( v(x) = f(x) \) with \( v'(x) = f'(x) \)Applying these, the derivative \( g'(x) \) becomes:\[ g'(x) = \frac{0 \cdot f(x) - 1 \cdot f'(x)}{(f(x))^2} = \frac{-f'(x)}{(f(x))^2}\]This shows how sensitive the reciprocal function is to changes in the original function.

Differentiable Function

A differentiable function is one that has a derivative at every point in its domain. In simpler terms, it means you can draw a tangent line at every point on its graph, without encountering any jumps, sharp corners, or discontinuities. Differentiability implies a smooth, continuous behavior.

• For a function to be differentiable, it must not have any points where the derivative does not exist.
• Differentiable functions also tend to be continuous; however, a continuous function may not necessarily be differentiable everywhere.
• In context of \( f(x) \) in our exercise, since it is differentiable, we can confidently find \( g'(x) \), the derivative of the reciprocal function, wherever \( f(x) eq 0 \).

Ensuring \( f(x) \) is differentiable allows us to use powerful calculus tools like the quotient rule to explore more complex functions like reciprocals.