Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(t)=\sqrt{5-t} $$

Short answer

The derivative of the function \(f(t)=\sqrt{5-t}\) is \(f'(t)= -\frac{1}{2\sqrt{5-t}}\).

Step-by-step solution

Identify the Inner Function and Outer Function

In this function \(f(t)=\sqrt{5-t}\), the inner function is \(u=5-t\) and the outer function is \(f(u)=\sqrt{u}\). Hence, according to the chain rule, the derivative is \(f'(t)=f'(u)u'\)

Find the Derivative of the Outer Function and Inner Function

Using the power rule, the derivative of the outer function is \(f'(u)=\frac{1}{2\sqrt{u}}\), and the derivative of the inner function is \(u'=-1\)

Substitute and Simplify

Substitute the values found in Step 2 into \(f'(t)=f'(u)u'\). So, \(f'(t)=\frac{1}{2\sqrt{u}}*(-1)= -\frac{1}{2\sqrt{5-t}}\). Here, we also substitute \(u\) with \(5-t\).

Key concepts

Understanding the Chain Rule

When dealing with composite functions, the chain rule is an essential technique for finding derivatives. A composite function is one that involves two or more functions combined in such a way that the output of one function becomes the input of another. In its simplest form, if you have a function that can be expressed as a composition of an outer function and an inner function, the chain rule helps you differentiate it efficiently. For example, suppose we have a function \( f(x) = g(h(x)) \). To find the derivative of \( f(x) \), we apply the chain rule, which states:

• First, take the derivative of the outer function \( g(u) \) with respect to its argument \( u \), keeping the inner function \( h(x) \) as it is.
• Then multiply it by the derivative of the inner function \( h(x) \) with respect to \( x \).

So in the mathematical form, it would be \( f'(x) = g'(h(x)) \cdot h'(x) \). By using the chain rule, we systematically solve for derivatives of nested functions, as seen in this problem where \( f(t) = \sqrt{5-t} \) requires differentiating both the inner and outer functions separately.

What is a Derivative?

Derivatives are a fundamental concept in calculus, expressing how a quantity changes with respect to another. Essentially, a derivative represents the rate of change or the slope of a function at any given point. Applying derivatives allows us to understand how variables transform and interact.

• The derivative of a function \( f(x) \) with respect to \( x \) is denoted \( f'(x) \).
• It is mathematically defined as the limit of the average rate of change of the function as the interval approaches zero. This can be symbolized as: \[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} \].

Understanding derivatives helps you solve problems involving rates, like speed, acceleration, and others. In finding the derivative of \( f(t) = \sqrt{5-t} \), recognizing the nature of derivatives ensures that we correctly apply rules like the chain rule and the power rule.

Utilizing the Power Rule

The power rule is a simple yet powerful tool for finding the derivative of polynomial functions or functions that can be expressed in terms of powers of \( x \). The rule states that for any real number \( n \), the derivative of \( x^n \) is \( n \cdot x^{n-1} \).

• This means that you "bring down" the exponent as a coefficient and then reduce the exponent by one.
• An example is differentiating \( x^3 \), which yields \( 3x^2 \).

In more complex derivatives, like with \( f(t) = \sqrt{5-t} \), you can still apply the power rule by rewriting the function in power form. Since \( \sqrt{u} \) is equivalent to \( u^{1/2} \), the derivative \( f'(u) = \frac{1}{2}u^{-1/2} \) demonstrates the power rule in action after identifying \( u=5-t \). Combining this with the chain rule reinforces how these mathematical concepts interact to solve problems efficiently.