Problem 3 - Calculus for Biology and Medicine

Question

Differentiate the functions given with respect to the independent variable. $$ f(x)=-2 x^{5}+7 x-4 $$

Step-by-Step Solution

Answer

The derivative is \(-10x^{4} + 7\).

Step 1: Understand Differentiation

Differentiation is finding the rate at which a function changes at any point. For a function given as a polynomial, you apply the power rule to each term separately.

Step 2: Apply the Power Rule

The power rule states that if you have a term of the form \( ax^n \), its derivative is \( a imes n imes x^{n-1} \).

Step 3: Differentiate Each Term

Apply the power rule to each term:1. \(-2x^5\) becomes \(-2 \times 5x^{5-1} = -10x^4\).2. \(7x\) becomes \(7 \times 1x^{1-1} = 7\).3. \(-4\) is a constant and becomes \(0\) when differentiated.

Step 4: Write the Overall Derivative

Combine all the derivatives of the terms to form the derivative of the function: \(-10x^4 + 7\).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for differentiating terms of a polynomial. Simply put, it allows us to efficiently find the derivative of any term in the form of \( ax^n \). Here's how it works:

• Take the exponent from \(x\), which is \(n\).
• Multiply the entire term by this exponent, effectively moving it in front of the term.
• Subtract one from the original exponent to get the new exponent.

For example, applying the power rule to \(-2x^5\), the exponent \(5\) is multiplied by \(-2\), resulting in \(-10\). You then reduce the original exponent by one, changing it from \(5\) to \(4\), yielding the term \(-10x^4\). This systematic approach simplifies polynomial differentiation significantly.

Polynomial Differentiation

Polynomial differentiation involves using calculus to find the derivative of polynomial functions. These functions are made up of several terms with variables raised to different powers. Each term of the polynomial can be differentiated individually using the power rule.

When faced with a polynomial like \(-2x^5 + 7x - 4\), you apply the power rule to each variable term separately. It's important to note that:

• Coefficients like \(7\) remain unchanged.
• Constant terms like \(-4\) vanish, since their derivative is zero.

Once you differentiate every term, you merge them to get the overall derivative of the polynomial. It's a step-by-step process but becomes intuitive with practice, making it easier to handle more complex polynomials.

Calculus Basics

Calculus is the branch of mathematics that deals with rates of change and accumulation. One of the foundations of calculus is differentiation, which helps us understand how a function changes with respect to its variables.

Key concepts in differentiation include:

• Derivatives represent the slope or rate of change.
• For polynomial functions, each individual term's rate of change can be found using the power rule.
• By looking at derivatives, we can gain insights into the behavior of curves, such as peaks and troughs.

Understanding these basics equips students with the ability to approach more advanced problems in calculus. Building a solid foundation allows for tackling complexities as you advance deeper into the study of calculus.