Problem 26 - Essential Calculus Early Transcendentals

Question

Differentiate the function. $$ y=10^{1-x^{2}} $$

Step-by-Step Solution

Answer

The derivative is \( y' = -2x \cdot 10^{1-x^2} \cdot \ln(10) \).

Step 1: Recognize the form

The function given is of the form \( y = a^{u(x)} \), where \( a = 10 \) and \( u(x) = 1-x^2 \). We will use the chain rule and the formula for differentiating exponential functions.

Step 2: Apply the differentiation formula for exponentials

The derivative of \( y = a^{u(x)} \) with respect to \( x \) is \( y' = a^{u(x)} \cdot \ln(a) \cdot u'(x) \). Here, \( a = 10 \), so we have \( y' = 10^{1-x^2} \cdot \ln(10) \cdot u'(x) \).

Step 3: Differentiate the inner function

Now, let's differentiate the inner function \( u(x) = 1 - x^2 \). The derivative \( u'(x) \) is \( -2x \).

Step 4: Substitute and simplify

Substituting \( u'(x) = -2x \) into the derived formula, we have \( y' = 10^{1-x^2} \cdot \ln(10) \cdot (-2x) \), which simplifies to \( y' = -2x \cdot 10^{1-x^2} \cdot \ln(10) \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus, especially useful when dealing with composite functions, where one function is nested inside another. In straightforward terms, the chain rule helps you differentiate a function by following a path from the outer layer to the innermost function.
For example, suppose you have a function like \(y = f(g(x))\). To differentiate this, you apply the chain rule by taking the derivative of the outer function \(f\) evaluated at the inner function \(g(x)\), and multiply it by the derivative of the inner function \(g(x)\).

• Identify the inner function \(g(x)\).
• Find its derivative \(g'(x)\).
• Differentiate the outer function \(f(u)\) with respect to \(u\), then multiply by \(g'(x)\).

This leapfrogging differentiation method will look like \(f'(g(x)) \cdot g'(x)\) in terms of the chain rule.

Exponential Functions

Exponential functions are a special family of functions that appear extensively in various areas of mathematics. An exponential function can be expressed as \(y = a^{u(x)}\), where \(a\) is a positive constant and \(u(x)\) is any function of \(x\).
Exponential functions have unique properties:

• They grow or decay at rates proportional to their own value.
• They have applications in natural processes and financial modeling.

For differentiation, the rule for an exponential function \(a^{u(x)}\) is \(\frac{d}{dx}[a^{u(x)}] = a^{u(x)} \cdot \ln(a) \cdot u'(x)\). This formula integrates both the exponential aspect \(a^{u(x)}\) and the natural logarithm \(\ln(a)\). It simplifies the process when combined with other calculus rules like the chain rule.

Derivative Computation

The computation of derivatives is a key task in calculus, which gives us information about the rate of change of a function. This process involves using known rules of differentiation, like the power rule, product rule, quotient rule, and chain rule.
In our problem, combining these methods is crucial:

• Recognize the form of the function, identifying components like exponential form.
• Use the derivative formula for that specific type, as seen with exponentials.
• Apply the chain rule if there is a composite function structure.
• Simplify your result by substituting back any derived expressions.

For the function \(y=10^{1-x^{2}}\), calculating the derivative involves pulling together these steps, tackling one piece of the function at a time until the full derivative is assembled.