Problem 1 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems 1-22 with respect to the independent variable. $$ f(x)=4 x^{3}-7 x+1 $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = 4x^3 - 7x + 1 \) is \( 12x^2 - 7 \).

Step 1: Differentiate term by term

To differentiate the function, differentiate each term separately based on differentiation rules. The function is given by \( f(x) = 4x^3 - 7x + 1 \).

Step 2: Differentiate the first term

The first term is \( 4x^3 \). Use the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \) to differentiate: \[ \frac{d}{dx}[4x^3] = 4 \cdot 3x^{3-1} = 12x^2 \]

Step 3: Differentiate the second term

The second term is \( -7x \). Again, use the power rule where the exponent is 1: \[ \frac{d}{dx} [-7x] = -7 \cdot 1x^{1-1} = -7 \]

Step 4: Differentiate the third term

The third term is a constant \( 1 \). The derivative of any constant is 0: \[ \frac{d}{dx}[1] = 0 \]

Step 5: Combine the derivatives of each term

Summing the derivatives from each term gives the overall derivative of the function: \[ \frac{d}{dx}[f(x)] = 12x^2 - 7 + 0 = 12x^2 - 7 \]

Key Concepts

Power Rule

Understanding differentiation can be quite exciting, especially with a simple rule like the Power Rule. This rule states that if you have a function of the form \( x^n \), then its derivative is given by \( nx^{n-1} \). Here's why it's handy:

• Applies to any function where a variable is raised to a power.
• Allows you to quickly find the rate of change of that function.
• Easy to use and remember, even if the exponent \( n \) is a fraction or a negative number.

To use this rule effectively, simply take the exponent, multiply it by the coefficient in front of the term, and then decrease the exponent by one. It's like taking a quick snapshot of how each part of the function behaves. In our example function \( 4x^3 - 7x + 1 \), the power rule was used to differentiate \( 4x^3 \), resulting in \( 12x^2 \). The process reveals how the function grows or shrinks as \( x \) changes.

Derivatives

Derivatives play a fundamental role in calculus and help us understand how a function changes. Think of a derivative as a tool that measures the sensitivity of a function's output to changes in its input. It's especially useful when:

• Determining the slope of a function at any given point.
• Finding the rate at which something changes, such as speed or growth.
• Identifying maximum and minimum points of a function, also known as critical points.

In our exercise, differentiating the function \( f(x)=4x^3-7x+1 \) involves finding the derivative for each term separately. The process is straightforward, and you can apply known rules like the power rule and constant term differentiation. By breaking down the function term by term and applying derivatives, you gain insights into how each component of the function contributes to its overall behavior and change.

Constant Term Differentiation

Constant term differentiation is perhaps the simplest rule when working with derivatives. It says that the derivative of any constant is always zero. Why? Because constants do not change, and thus their rate of change is zero. In simple terms:

• A constant remains fixed, irrespective of the variable changes.
• This makes constant terms disappear in the derivative expression since their contribution to change is null.
• Simplifies calculations when dealing with polynomials or functions combining multiple terms.

In our function \( f(x) = 4x^3 - 7x + 1 \), the constant term \( 1 \) becomes \( 0 \) after differentiation. This clearly demonstrates how constants have no effect on the rate at which a function changes. Remembering this can help streamline your differentiation process, leading to quicker and more efficient problem-solving.