Question

Compute the derivative of the given function. $$f(t)=\sqrt{1-t^{2}}$$

Short answer

The derivative is \( f'(t) = -\frac{t}{\sqrt{1-t^2}} \).

Step-by-step solution

Rewrite the Function

Convert the square root into an exponent notation to simplify differentiation. The function can be rewritten as \( f(t) = (1-t^2)^{1/2} \).

Apply the Chain Rule

To differentiate \( f(t) = (1-t^2)^{1/2} \), use the chain rule. The chain rule states that if you have a composite function \( (g(h(t))) \), then its derivative is \( g'(h(t)) \cdot h'(t) \).

Differentiate the Outer Function

Differentiate the outer function \( g(u) = u^{1/2} \). Using the power rule, the derivative is \( g'(u) = \frac{1}{2}u^{-1/2} \).

Differentiate the Inner Function

Identify the inner function \( h(t) = 1-t^2 \). The derivative is \( h'(t) = -2t \) since it is a simple polynomial.

Combine Using the Chain Rule

Combining the results, the derivative \( f'(t) \) is \( \frac{1}{2}(1-t^2)^{-1/2} \cdot (-2t) \).

Simplify the Derivative

Simplify the expression obtained in the previous step: \( f'(t) = -\frac{t}{\sqrt{1-t^2}} \).

Key concepts

Chain Rule

The chain rule is a fundamental rule in calculus for differentiating composite functions. It helps us determine the derivative of a function that is composed of two or more functions. Imagine you have an outer function that wraps around one or more inner functions, just like an onion with its layers. The chain rule allows you to peel back these layers, one by one, to find the derivative.
This concept is especially useful when dealing with complex functions that aren't just simple polynomials.In the exercise above, the function provided is a composite function, written as \( f(t) = (1-t^2)^{1/2} \). Here, the chain rule is used to differentiate the outer function \( (g(u) = u^{1/2}) \) and the inner function \( (h(t) = 1-t^2) \). Following the chain rule formula \( g'(h(t)) \cdot h'(t) \), you can see how it becomes easier to tackle the problem, even if the function itself seems a bit complicated at first.Using the chain rule effectively requires you to correctly identify the inner and outer parts of your composite function and to differentiate each accordingly.

Power Rule

The power rule is a quick and handy differentiation rule that is incredibly useful when dealing with exponents. It's a tool that tells you how to differentiate a function that has the form of a variable raised to a power. The power rule states that if you have a term like \( x^n \), then its derivative is \( nx^{n-1} \). This is a straightforward way to handle not only simple power functions but also more complex ones when combined with other rules like the chain rule.In our exercise solution, the power rule is used to differentiate the outer function, which is written as \( g(u) = u^{1/2} \). By applying the power rule here, we find that the derivative is \( g'(u) = \frac{1}{2}u^{-1/2} \). This step is the application of the power rule,"s covering the simplification of the problem and is a key part of obtaining the final solution.

Differentiation

Differentiation is one of the basic concepts in calculus that forms the backbone of more complex operations. It is the mathematical process of finding a derivative, which represents how a function changes as its input changes. In simple terms, it's about finding the rate at which something changes at any point.When we differentiate a function like \( f(t)=\sqrt{1-t^{2}} \), we want to find out how this function behaves as \( t \) varies. This gives us the derivative of the function, denoting the rate of change of the function concerning \( t \). Differentiation is crucial when you need to analyze problems involving rates and slopes, whether in physics, economics, or engineering.In the provided textbook solution, differentiation involves using both the chain rule and the power rule to effectively manage the composite nature of the problem. This dual approach allows us to rigorously calculate the derivative, which is ultimately simplified to \( f'(t) = -\frac{t}{\sqrt{1-t^2}} \). This result tells us exactly how the original function changes at any given point \( t \).

Composite Function

A composite function arises when one function is applied inside another function. It can be understood as a function that 'feeds into' another, creating a distinct chain of dependency. This notion is essential, especially when we deal with functions that are not direct or simple.Looking back at our function \( f(t) = \sqrt{1-t^2} \), we see that it is comprised of two distinct parts: an inside function and an outside function. Here, \( 1-t^2 \) is the inner function and \( \sqrt{u} \) (or \( u^{1/2} \) in exponential form) is the outer function. By interpreting such a problem using the concept of composite functions, you can then apply the chain rule for differentiation.Composite functions appear in various real-world applications where phenomena can be broken down into simpler equations within larger processes. Recognizing when you are dealing with a composite function is crucial for selecting the appropriate differentiation method and simplifying the potentially complex operations of calculus.