Question

Compute the derivative of the given function. $$f(x)=\left(16 x^{3}+24 x^{2}+3 x\right) \frac{7 x-1}{16 x^{3}+24 x^{2}+3 x}$$

Short answer

The derivative is 7.

Step-by-step solution

Simplify the Function

The given function is \[ f(x) = \left(16x^3 + 24x^2 + 3x\right) \cdot \frac{7x - 1}{16x^3 + 24x^2 + 3x} \]Notice that the first term \( (16x^3 + 24x^2 + 3x) \) cancels out the same term in the denominator. So, the function simplifies to \( f(x) = 7x - 1 \).

Compute the Derivative

Since the simplified function is \( f(x) = 7x - 1 \), we need to find its derivative. Recall that the derivative of \( ax + b \) is simply \( a \).

Apply the Derivative Formula

For the function \( f(x) = 7x - 1 \), the derivative \( f'(x) \) is \( 7 \) because the derivative of a constant is zero and the derivative of \( 7x \) is \( 7 \).

Conclude with the Derivative

The derivative of the function \( f(x) = 7x - 1 \) is simply \( f'(x) = 7 \).

Key concepts

Simplification of Functions

Simplifying functions is a key step in solving many calculus problems. It involves reducing a complex expression into a simpler form that is easier to work with. In the given exercise, the function is \[ f(x) = \left(16x^3 + 24x^2 + 3x\right) \cdot \frac{7x - 1}{16x^3 + 24x^2 + 3x} \]which initially looks intimidating. However, by examining the function, you'll note that the numerator and the denominator consist of the same expression, \( 16x^3 + 24x^2 + 3x \). When these are in a division, they cancel each other out, which greatly simplifies the function to \( f(x) = 7x - 1 \).This step of simplification is crucial because it transforms our function into a simple linear function. Translation: we reduced a potentially complicated task into something more manageable. Always look for opportunities to simplify functions before proceeding with further calculations. It saves time and reduces errors.

Basic Differentiation

Differentiation is one of the fundamental concepts in calculus. It's the process by which you determine the rate at which a function is changing at any point. For basic differentiation of a simple linear function such as \( f(x) = ax + b \), the rule is straightforward: the derivative is simply the constant \( a \). This is because differentiation measures the slope of the function, and in a linear function, this slope is constant.In our simplified function \( f(x) = 7x - 1 \), the derivative is found by looking at the coefficient of \( x \). Here, \( a = 7 \), so the derivative of this linear function is \( f'(x) = 7 \). Remember, any constant added or subtracted in the function (like \( -1 \) here) will have a derivative of zero. Thus, basic differentiation helps us quickly find how a function behaves without constructing elaborate calculations.

Calculus Problems

Calculus problems often require multiple steps and use a variety of techniques. Whether it's finding limits, creating integrals, or calculating derivatives, a structured approach is essential. In our exercise, the problem asked us to find the derivative, but first required simplification. Issues that can arise in calculus problems include:

• Complex expressions: Simplification is key to managing these.
• Multiple steps: Frequently, problems include simplification, differentiation, or integration.
• Attention to detail: Missing a step or overlooking a cancellation can lead to incorrect answers.

For effective problem-solving: - Start by simplifying the problem if possible. - Carefully apply known formulas or rules of calculus. - Check each step for accuracy to ensure the final solution is correct. Calculus problems, once mastered, reveal fascinating insights into the world of change and motion.