Problem 37 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \underline{\phantom{xxx}} \begin{aligned} f(x)=x^{5} & \\ &\left[\text { Hint: Use } \quad(x+h)^{5}=\right.\\\ &\left.x^{5}+5 x^{4} h+10 x^{3} h^{2}+10 x^{2} h^{3}+5 x h^{4}+h^{5}\right] \end{aligned} $$

Step-by-Step Solution

Answer

The derivative \( f'(x) \) is \( 5x^4 \).

Step 1: Write the definition of the derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). Our goal is to find this expression for \( f(x) = x^5 \).

Step 2: Substitute \( f(x) \) and \( f(x+h) \)

Using the hint provided, express \( f(x+h) \) as \( (x+h)^5 = x^5 + 5x^4 h + 10x^3 h^2 + 10x^2 h^3 + 5x h^4 + h^5 \). So, we have:\[ f(x+h) - f(x) = (x^5 + 5x^4 h + 10x^3 h^2 + 10x^2 h^3 + 5x h^4 + h^5) - x^5 \].

Step 3: Simplify the expression

Simplify \( f(x+h) - f(x) \) by canceling \( x^5 \) terms:\[ 5x^4 h + 10x^3 h^2 + 10x^2 h^3 + 5x h^4 + h^5 \].

Step 4: Factor out \( h \) from each term

Factor \( h \) from the expression obtained:\[ h(5x^4 + 10x^3 h + 10x^2 h^2 + 5x h^3 + h^4) \].

Step 5: Divide by \( h \) and take the limit

Divide the factored expression by \( h \) and take the limit as \( h \to 0 \):\[ \lim_{h \to 0} (5x^4 + 10x^3 h + 10x^2 h^2 + 5x h^3 + h^4) = 5x^4 \].

Step 6: Conclusion of the calculation

The derivative \( f'(x) = 5x^4 \) is found after canceling terms that become \( 0 \) when \( h \to 0 \).

Key Concepts

Definition of Derivative

The derivative of a function is a foundational concept in calculus. It represents the rate at which a function changes as its input changes. You can think of the derivative as a formula that tells us the slope of the tangent line to the graph of the function at any point. This slope indicates how steep the graph is at that particular point.

To formally define the derivative of a function \(f(x)\), we use the limit process which gives us \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). This means we look at how the function's output changes as we change the input by a very small amount \(h\), and then see what happens as this change \(h\) approaches zero. The existence of this limit proves the function is differentiable at that point.

Limit Process

The limit process is a critical component in finding derivatives. It involves evaluating the behavior of a function as its variable approaches a particular value. In the case of finding derivatives, we use the notation \(\lim_{h \to 0}\). This signifies looking at what happens as \(h\), the small change in input, gets closer and closer to zero.

Let's break it down step by step:

• First, consider the function at slightly shifted point: \(f(x+h)\).
• Next, subtract the original function: \(f(x+h) - f(x)\). This difference tells us how much the function has "risen" over the small increment \(h\).
• Now, divide by \(h\) to find the average rate of change over the interval \([x, x+h]\).
• Finally, take the limit as \(h\) approaches zero to find the precise instantaneous rate of change, or the derivative.

This process helps us capture the essence of change at an exact point, making it an invaluable tool in calculus.

Polynomial Differentiation

Polynomial differentiation is a process of finding the derivative of polynomial functions. Polynomials are expressions consisting of variables raised to whole number powers, like \(x^5\). The power rule is a simple method used in calculus for differentiating such functions. It states that for a term of the form \(x^n\), its derivative is \(nx^{n-1}\).

In our specific example with \(f(x) = x^5\), applying the power rule directly would give \(f'(x) = 5x^4\). This is consistent with the derivative we found using the definition of derivative and limit process.

• Each term in the polynomial can be differentiated separately using the power rule.
• After differentiation, combine the results to get the derivative of the entire polynomial.

This method streamlines the process significantly for polynomial functions and allows for quick and efficient calculation of derivatives.