Problem 25 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=2 \tan \left(1-x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -4x \sec^2(1-x^2) \).

Step 1: Identify the Function

The function given is \( f(x) = 2 \tan(1-x^2) \). We are tasked with finding its derivative with respect to \( x \).

Step 2: Apply the Chain Rule

The function is a composite function. Start by identifying the outer function as \( 2 \tan(u) \) where \( u = 1-x^2 \). The derivative of \( \tan(u) \) with respect to \( u \) is \( \sec^2(u) \). So, the derivative of \( 2 \tan(u) \) with respect to \( u \) is \( 2 \sec^2(u) \).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \( u = 1 - x^2 \) with respect to \( x \). The derivative of \( 1 - x^2 \) is \( -2x \).

Step 4: Combine the Derivatives

Utilize the chain rule to combine the derivatives: \[ f'(x) = 2 \sec^2(1-x^2) \times (-2x) \]. Simplifying, we find: \[ f'(x) = -4x \sec^2(1-x^2) \].

Step 5: Provide the Final Answer

The derivative of the function \( f(x) = 2 \tan(1-x^2) \) with respect to \( x \) is \( f'(x) = -4x \sec^2(1-x^2) \).

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental tool for finding the derivative of a composite function. A composite function is essentially a function within another function, like layering a cake. When we have a composite function, we apply the chain rule to differentiate it.

• The chain rule states that if you have a composite function, the derivative of the outer function is multiplied by the derivative of the inner function.
• In mathematical terms, if you have functions \( f(g(x)) \), the derivative \( f'(x) \) is found by multiplying the derivative of \( f \) at \( g(x) \) by the derivative of \( g(x) \).

To clearly understand the application, consider our exercise: the function \( f(x) = 2 \tan(1-x^2) \). Here the outer function is \( 2 \tan(u) \) and the inner function is \( u = 1 - x^2 \). By applying the chain rule, we see that we first differentiate the outer function and then multiply it by the derivative of the inner function. This step gives depth to our calculation and ensures our derivatives are accurately solved.

Composite Function

A composite function is like a nested package – one function inside another. This is typical when one function's output becomes the input for another.

• For a function \( f(g(x)) \), \( g(x) \) is the inner function, and \( f \) is the outer function.
• This setup allows us to efficiently handle changes in variables affecting the overall output of a combination of functions.

In the original exercise, the function \( f(x) = 2 \tan(1-x^2) \) is a composite function where \( 1-x^2 \) acts as the inner part. This inner portion influences the evaluation of the tangent function. Recognizing and handling composite functions with care allows us to correctly apply the chain rule, breaking down complex derivatives into manageable pieces. In doing so, we ensure each function's role is clear and contributes accurately to the final derivative.

Trigonometric Derivative

Trigonometric derivatives focus on the rates of change for trigonometric functions like sine, cosine, and tangent. Understanding these is crucial if trigonometric expressions appear in your calculations.

• The derivative of \( \tan(u) \) is \( \sec^2(u) \). This means that when \( u \) changes slightly, the change in \( \tan(u) \) is affected significantly, emphasized by the \( \sec^2 \) term.
• Mastering these standard derivatives allows us to navigate more complex problems involving trigonometric functions.

In the problem at hand, the function uses the tangent derivative specifically. When dealing with \( 2 \tan(1-x^2) \), you first focus on the \( \tan(u) \) part. Its derivative \( 2 \sec^2(u) \) becomes vital. This helps express how changes in \( x \) affect the tangent's behavior, further altered by subsequent computations of the inner function. Being familiar with trigonometric derivatives makes handling these mathematical challenges manageable and strategic.