Problem 19 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=20 x^{3}-4 x^{6}+9 x^{8} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 60x^2 - 24x^5 + 72x^7 \).

Step 1: Apply the Power Rule

To differentiate each term of the function \( f(x) = 20x^3 - 4x^6 + 9x^8 \), we use the power rule, which states \( \frac{d}{dx} [x^n] = n \cdot x^{n-1} \). We will differentiate each term individually.

Step 2: Differentiate the First Term

The first term is \( 20x^3 \). Applying the power rule: \( \frac{d}{dx} [20x^3] = 3 \cdot 20 \cdot x^{3-1} = 60x^2 \).

Step 3: Differentiate the Second Term

The second term is \( -4x^6 \). Applying the power rule: \( \frac{d}{dx} [-4x^6] = 6 \cdot (-4) \cdot x^{6-1} = -24x^5 \).

Step 4: Differentiate the Third Term

The third term is \( 9x^8 \). Applying the power rule: \( \frac{d}{dx} [9x^8] = 8 \cdot 9 \cdot x^{8-1} = 72x^7 \).

Step 5: Combine the Derivatives

Now, we combine the derivatives of each term to find \( f'(x) \): \( f'(x) = 60x^2 - 24x^5 + 72x^7 \).

Key Concepts

Power Rule

The power rule is a basic yet powerful tool in calculus, especially when dealing with simple polynomial functions. It simplifies the process of finding derivatives, which represent the rate of change of a function. The rule is straightforward: if you have a function of the form \( f(x) = x^n \), the derivative \( f'(x) \) is calculated as \( n \cdot x^{n-1} \). This means you multiply the original exponent by the coefficient and then reduce the exponent by one.For example, in applying the power rule to a term like \( 20x^3 \):

• Multiply the exponent (3) by the coefficient (20), resulting in 60.
• Subtract 1 from the exponent, changing it from 3 to 2.

Thus, the derivative of \( 20x^3 \) is \( 60x^2 \).This rule makes differentiation straightforward, particularly when dealing with polynomial functions, enabling you to handle multiple terms by working through them one by one.

Derivative

A derivative essentially describes how a function changes as its input changes. It's the backbone of differential calculus, providing insights into functions' behaviors, such as where they increase, decrease, or have unique points like local maxima or minima.The derivative of a function \( f(x) \) at any point tells you the slope of the tangent line to the graph at that point. If \( f'(x) \) is positive, the function is increasing at that point; if negative, it’s decreasing.Differentiating a polynomial function like \( f(x) = 20x^3 - 4x^6 + 9x^8 \) involves:

• Applying the power rule to each term to calculate \( f'(x) \).
• Understanding the physical meaning of the resulting expression \( f'(x) = 60x^2 - 24x^5 + 72x^7 \).

The function's derivative provides a new function, giving the slope for any input \( x \). This makes derivatives a crucial tool for examining and predicting how functions behave over intervals.

Polynomial Differentiation

Polynomial differentiation is the process of finding the derivative of a polynomial function, which can have multiple terms, each involving a power of \( x \).A polynomial function, like \( f(x) = 20x^3 - 4x^6 + 9x^8 \), contains terms with different exponents and coefficients. Differentiating such a function involves using the power rule repeatedly:

• Work through each term individually.
• Combine all derived terms to form the full derivative.

Let's differentiate the given polynomial:

• The derivative of \( 20x^3 \) is \( 60x^2 \) using the power rule.
• The derivative of \( -4x^6 \) becomes \( -24x^5 \).
• Finally, \( 9x^8 \) yields a derivative of \( 72x^7 \).

These derivatives sum to give the final answer: \( f'(x) = 60x^2 - 24x^5 + 72x^7 \).Polynomial differentiation is not only a methodical approach but also a foundation for more advanced mathematics. As you become more familiar with these techniques, you'll see how they simplify complex real-world problems.