Solving A Differentiable Functions Problem Finding 3g'(2) + 2f'(2)

In the realm of calculus, differentiable functions play a pivotal role, allowing us to analyze the rates at which functions change. This article delves into a fascinating problem involving two differentiable functions, f and g, and seeks to determine the value of the expression 3g'(2) + 2f'(2). We are given that g(2) = 4 and f(1/x) = x g(x + x^2). Let's embark on a journey to unravel this mathematical puzzle, employing the principles of calculus and differentiation.

We are presented with a scenario where f and g are differentiable functions defined over the real numbers. Our goal is to find the value of 3g'(2) + 2f'(2), given the functional relationship f(1/x) = x g(x + x^2) and the specific value g(2) = 4. This problem skillfully combines function composition, differentiation, and the evaluation of derivatives at specific points.

To tackle this problem effectively, we'll need to employ the chain rule and product rule of differentiation. The chain rule will be crucial for differentiating composite functions like f(1/x), while the product rule will be necessary for differentiating terms like x g(x + x^2). The given functional relationship connects the functions f and g, and by differentiating both sides, we can establish a relationship between their derivatives, f ' and g'.

The heart of the solution lies in differentiating the equation f(1/x) = x g(x + x^2) with respect to x. Let's break down this process step by step. On the left-hand side, we have a composite function, so we'll apply the chain rule. The derivative of f(1/x) with respect to x is f '(1/x) multiplied by the derivative of 1/x with respect to x. The derivative of 1/x is -1/x^2. Therefore, the derivative of the left-hand side is f '(1/x) * (-1/x^2).

Now, let's turn our attention to the right-hand side, x g(x + x^2). Here, we have a product of two functions, so we'll use the product rule. The product rule states that the derivative of uv with respect to x is u'v + uv' . In our case, u = x and v = g(x + x^2). The derivative of u with respect to x is simply 1. The derivative of v with respect to x requires the chain rule, as we have a composite function. The derivative of g(x + x^2) with respect to x is g '(x + x^2) multiplied by the derivative of (x + x^2) with respect to x, which is (1 + 2x). Thus, the derivative of the right-hand side is 1 * g(x + x^2) + x * g '(x + x^2) * (1 + 2x).

Equating the derivatives of both sides, we obtain the equation: f '(1/x) * (-1/x^2) = g(x + x^2) + x * g '(x + x^2) * (1 + 2x). This equation establishes a crucial relationship between f ' and g ', which we will use to solve for the desired expression.

Our goal is to find the value of 3g'(2) + 2f'(2). We have an equation relating f '(1/x) and g '(x + x^2). To make use of this equation, we need to choose a value for x that will allow us to evaluate f '(2) and g'(2). Looking at the arguments of the derivative functions, we see that if we set 1/x = 2, then x = 1/2, and we can evaluate f '(2). Furthermore, with x = 1/2, the argument of g ' becomes x + x^2 = (1/2) + (1/4) = 3/4. This doesn't directly give us g'(2). To directly obtain g'(2), we need to find an x such that x + x^2 = 2. This quadratic equation can be rewritten as x^2 + x - 2 = 0, which factors into (x + 2)(x - 1) = 0. The solutions are x = 1 and x = -2.

Let's analyze each of these potential x values. If x = 1, then 1/x = 1, which means we'll be evaluating f '(1), not f '(2). However, if x = -2, then 1/x = -1/2, so we'll be evaluating f '(-1/2), and x + x^2 = -2 + 4 = 2, which is exactly what we need to evaluate g'(2). Therefore, the appropriate value to substitute into our equation is x = -2.

Substituting x = -2 into the equation f '(1/x) * (-1/x^2) = g(x + x^2) + x * g '(x + x^2) * (1 + 2x), we get: f '(-1/2) * (-1/4) = g(2) + (-2) * g '(2) * (1 + 2(-2)).

We know that g(2) = 4, so we can substitute that value into the equation: f '(-1/2) * (-1/4) = 4 + (-2) * g '(2) * (-3). This simplifies to: f '(-1/2) * (-1/4) = 4 + 6g'(2).

Multiplying both sides by -4, we obtain: f '(-1/2) = -16 - 24g'(2). This equation relates f '(-1/2) and g'(2), but it doesn't directly involve f '(2), which we need to find the value of 3g'(2) + 2f'(2).

Since setting 1/x = 2 gives us x = 1/2, let's substitute x = 1/2 into the differentiated equation: f '(1/x) * (-1/x^2) = g(x + x^2) + x * g '(x + x^2) * (1 + 2x).

This yields: f '(2) * (-1/(1/4)) = g(1/2 + 1/4) + (1/2) * g '(1/2 + 1/4) * (1 + 2(1/2)). Simplifying, we get: f '(2) * (-4) = g(3/4) + (1/2) * g '(3/4) * 2. Further simplification gives: -4f '(2) = g(3/4) + g '(3/4).

This equation involves f '(2), but it also includes g(3/4) and g '(3/4), which we don't know. So, this equation alone doesn't provide a direct solution. We need to revisit our strategy.

We have two equations now:

1. f '(-1/2) = -16 - 24g'(2)
2. -4f '(2) = g(3/4) + g '(3/4)

Neither of these equations directly gives us the value of 3g'(2) + 2f'(2). We need to find a way to connect f '(2) and g'(2) more directly. Let's return to the original differentiated equation and consider a different substitution strategy.

We have: f '(1/x) * (-1/x^2) = g(x + x^2) + x * g '(x + x^2) * (1 + 2x).

To find f '(2), we need 1/x = 2, which means x = 1/2. We already used this substitution, but let's examine the result more closely:

-4f '(2) = g(3/4) + g '(3/4). This equation has f '(2) but also involves g(3/4) and g '(3/4), making it difficult to solve directly.

To find g'(2), we need x + x^2 = 2, which leads to x^2 + x - 2 = 0. This factors into (x + 2)(x - 1) = 0, giving us x = 1 or x = -2. We already used x = -2. Let's try x = 1.

Substituting x = 1 into the differentiated equation, we get: f '(1) * (-1) = g(2) + 1 * g '(2) * (1 + 2). Since g(2) = 4, we have: -f '(1) = 4 + 3g'(2).

This equation relates f '(1) and g'(2), but it doesn't directly help us find f '(2).

We've explored several avenues, and now it's time to synthesize our findings. We have the following key equations:

1. f '(1/x) * (-1/x^2) = g(x + x^2) + x * g '(x + x^2) * (1 + 2x)
2. Substituting x = -2: f '(-1/2) * (-1/4) = 4 + 6g'(2), which simplifies to f '(-1/2) = -16 - 24g'(2)
3. Substituting x = 1/2: -4f '(2) = g(3/4) + g '(3/4)

Let’s reconsider the substitution that led us to g’(2). We know by substituting x=-2: f '(-1/2) * (-1/4) = g(2) + (-2) * g '(2) * (1 + 2(-2)) f '(-1/2) * (-1/4) = 4 + 6g'(2) f '(-1/2) = -16 - 24g'(2)

Also, let’s reconsider the x=1 case: -f '(1) = 4 + 3g'(2) f '(1) = -4 - 3g'(2)

Unfortunately, we still don't have f'(2).

Let’s review our goal again, which is 3g'(2) + 2f'(2). We can manipulate these equations to find 3g'(2) + 2f'(2). Let's go back to f '(1/x) * (-1/x^2) = g(x + x^2) + x * g '(x + x^2) * (1 + 2x).

If we look closer, the substitution x=1/2 gave us: -4f '(2) = g(3/4) + g '(3/4)

This doesn't directly involve g'(2).

But let’s try to make a different substitution. Let’s try implicit differentiation directly without further substitutions. Let u = 1/x, then x = 1/u f(u) = (1/u) * g(1/u + 1/u^2)

Take the derivative with respect to u: f '(u) = (-1/u^2) * g(1/u + 1/u^2) + (1/u) * g' (1/u + 1/u^2) * (-1/u^2 - 2/u^3)

Now let u = 2: f '(2) = (-1/4) * g(1/2 + 1/4) + (1/2) * g' (1/2 + 1/4) * (-1/4 - 2/8) f '(2) = (-1/4) * g(3/4) + (1/2) * g' (3/4) * (-1/2) f '(2) = (-1/4) * g(3/4) - (1/4) * g' (3/4) -4 f '(2) = g(3/4) + g' (3/4)

This is equivalent to our earlier substitution x=1/2.

From here, it's difficult to make further progress without further information or a clever trick we are missing. Therefore, after reviewing the steps, let's examine our original working with x = -2: f '(-1/2) = -16 - 24g'(2) and consider if there is a calculation error in the options or the question. Based on our calculations we believe the provided answer choices are incorrect.

Through a rigorous application of differentiation rules and strategic substitutions, we have explored the intricate relationship between the derivatives of the given functions. However, despite our efforts, we encountered an impasse in finding a definitive numerical value for 3g'(2) + 2f'(2) using the information provided. This underscores the importance of careful analysis, strategic problem-solving, and the recognition of limitations in mathematical problem-solving.

Keywords: differentiable functions, chain rule, product rule, differentiation, derivatives, function composition, calculus, mathematics.