Problem 9 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the derivative of the function. $$ f(x)=e^{-1 / x^{2}} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = e^{-1 / x^{2}}\) is \(f'(x) = 2e^{-1 / x^{2}} / x^{3}\).

Step 1: Identify the Inner Function

Express the given function as \(e^{u(x)}\), where \(u(x) = -1 / x^{2}\). This is called the 'inner function'.

Step 2: Take Derivative of the Inner Function

Then, differentiate \(u(x) = -1 / x^{2}\) with respect to \(x\) using the power rule, obtaining \(u'(x) = 2 / x^{3}\).

Step 3: Apply the Chain Rule

Now that we know both the outer function and the derivative of the inner function, we can apply the Chain Rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function, leading finally to \(f'(x) = e^{u(x)} * u'(x)\). Substituting in our values we have calculated earlier, \(f'(x) = e^{-1 / x^{2}} * 2 / x^{3}\).

Key Concepts

Chain Rule

When it comes to finding the derivative of a composite function, the chain rule is a powerful technique. The chain rule helps us differentiate functions within functions, often denoted as outer and inner functions.

A common example is when you have a function with another function inside it, like \( f(x) = e^{-1/x^2} \). Here, the outer function is the exponential function, and the inner function is \( u(x) = -1/x^2 \).

• The chain rule formula is: \( f'(x) = g'(h(x)) \cdot h'(x) \),
• where \( g(h(x)) \) is your outer function, and \( h(x) \) is your inner function.

In the solution provided, the inner function was differentiated first, using the power rule, yielding \( u'(x) = 2 / x^3 \).
Then, the chain rule combines the derivatives of the inner function with the outer function's derivative. Hence, the formula \( f'(x) = e^{-1/x^2} \cdot (2/x^3) \) gives the complete derivative of the composite function.

Power Rule

The power rule is an essential part of calculus that helps us find the derivative of functions raised to a power. It's a straightforward rule that says if you have a function \( x^n \), then its derivative is \( nx^{n-1} \).

• This rule applies to any integer power \( n \) and is very useful when differentiating polynomials or parts of composite functions.

For instance, in the inner function \( u(x) = -1/x^2 \), it can be rewritten as \( u(x) = -x^{-2} \).
By applying the power rule, we find \( u'(x) = -(-2)x^{-2-1} = 2/x^3 \).
This use of the power rule is crucial before applying the chain rule, as it allows us to handle the derivative of the inner function efficiently. Remember, identifying and differentiating inner components with the power rule simplifies the process.

Composite Function

A composite function is like a set of nested functions, where you plug one function into another. In a composite function like \( f(x) = e^{-1/x^2} \), the composition happens because \( -1/x^2 \) is inside the exponential function.

Recognizing a composite function is the first step to finding its derivative using rules like the chain and power rules. Composite functions can be a bit tricky because they involve more than one type of operation. However, breaking them down into inner and outer functions makes differentiation manageable. You identify these two parts to correctly apply the chain rule's methodology.

• Composite functions are prevalent and allow greater flexibility when forming mathematical models.
• Knowing how to differentiate them opens up many doors in calculations and understanding function behavior.

With practice, solving derivative problems using composite functions becomes more intuitive.