Problem 6 - Calculus for Biology and Medicine

Question

Differentiate the functions given with respect to the independent variable. $$ f(x)=-1+3 x^{2}-2 x^{4} $$

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = 6x - 8x^3 \).

Step 1: Understand the Problem

We are given the function \( f(x) = -1 + 3x^2 - 2x^4 \) and are asked to find its derivative with respect to \(x\). Differentiating involves finding the rate of change of the function with respect to \(x\).

Step 2: Apply Power Rule for Differentiation

The power rule for differentiation states that \(\frac{d}{dx}[x^n] = nx^{n-1}\). We will apply this rule to each term in the function \( f(x) = -1 + 3x^2 - 2x^4 \).

Step 3: Differentiate Each Term in the Function

- The derivative of the constant \(-1\) is \(0\). - For the term \(3x^2\), using the power rule, we get \(\frac{d}{dx}[3x^2] = 3 \cdot 2x^{2-1} = 6x\). - For the term \(-2x^4\), using the power rule, we get \(\frac{d}{dx}[-2x^4] = -2 \cdot 4x^{4-1} = -8x^3\).

Step 4: Combine the Derivatives

Combine the derivatives of each term to find the overall derivative of the function: \( \frac{d}{dx}[-1 + 3x^2 - 2x^4] = 0 + 6x - 8x^3 \). Thus, the derivative of the function is \( f'(x) = 6x - 8x^3 \).

Key Concepts

Power Rule

When we differentiate functions composed of polynomial terms, one of our most vital tools is the power rule. The power rule simplifies the process of finding derivatives significantly.
It states that if you have a term in the form of \[x^n\], its derivative is \[nx^{n-1}\]. This means you take the exponent of the variable, multiply it by the coefficient of the term, and then reduce the exponent by one.Using the power rule allows you to swiftly handle differentiation of terms like \[3x^2\] and \[-2x^4\].

• For \[3x^2\], apply the power rule: multiply by the exponent (2) and decrease the exponent by one, giving \[6x\].
• Likewise, for \[-2x^4\], multiply by the exponent (4) yielding \[-8x^3\].

Remember, constants are treated differently while differentiating – the derivative of a constant is simply zero.

Derivative

The derivative of a function gives you a new function which represents the rate at which the original function changes.
This is often interpreted as the slope of the tangent line to the curve at any point.When differentiating, you are essentially measuring how fast or slow the function is rising or falling with respect to the independent variable, usually denoted as \(x\). In simple terms:

• The derivative tells us how steep the original function is at any given point.
• Where the derivative is positive, the function is increasing.
• Where it is negative, the function is decreasing.

In our example, the function \(f(x) = -1 + 3x^2 - 2x^4\) transforms to its derivative \(f'(x) = 6x - 8x^3\). This derivative gives insight into how our original function behaves, including whether it is rising or falling at specific values of \(x\).

Function

Understanding the concept of a function is crucial in topics like differentiation. A function in mathematics is a relation that uniquely associates elements of one set with elements of another set. Typically, we deal with functions of the form \(f(x)\), which signify how outputs are determined based on inputs.The function \(f(x) = -1 + 3x^2 - 2x^4\) is a polynomial function and can be visualized as a curve on a graph.
Each term in this function, like \(-1\), \(3x^2\), and \(-2x^4\), contributes to the overall shape of the graph.

• The constant term \(-1\) provides the "vertical shift."
• The term \(3x^2\) shapes the parabola, causing it to open upward.
• The term \(-2x^4\) influences the rate at which the curve grows.

Understanding this, differentiating the function simply gives a new perspective to interpret the behavior of the curve through its rate of changes and slopes.