what is the answer?---**Question Number:** 12 **Question Stem:** Tilde deriverar funktionen f(x) = e^(2x) och ställer upp kvoten f'(x) / f(x). Hon påstår följande: "För alla värden på x kommer kvoten alltid att få värdet 2". Har Tilde rätt? Motivera ditt svar. **Other Relevant Text:** (1/0/0)

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In this problem, Tilde claims that for the function f of x equals e to the power of 2x, the quotient of the derivative f prime of x divided by the function f of x always equals 2 for all values of x. Let's verify if her claim is correct. First, we need to find the derivative of f of x equals e to the power of 2x. Using the chain rule, the derivative of e to the power of u is e to the power of u times the derivative of u. Here, u equals 2x, so u prime equals 2. Therefore, f prime of x equals e to the power of 2x times 2, which is 2e to the power of 2x. Next, we calculate the quotient f prime of x divided by f of x. This gives us 2e to the power of 2x divided by e to the power of 2x. Since e to the power of 2x is never zero for any real value of x, we can cancel out this term from the numerator and denominator. The simplified quotient equals 2. Therefore, Tilde's claim is correct: the quotient f prime of x divided by f of x equals 2 for all values of x. Let's understand why the quotient f prime of x divided by f of x equals 2 for the function f of x equals e to the power of 2x. This is actually part of a general pattern for exponential functions. For any function of the form f of x equals e to the power of ax, where a is a constant, the derivative is f prime of x equals a times e to the power of ax. When we calculate the quotient f prime of x divided by f of x, we get a times e to the power of ax divided by e to the power of ax, which simplifies to just a. This pattern applies to all exponential functions where the base is e and the exponent is a linear function of x. In our specific case, a equals 2, so the quotient equals 2. We can see this pattern with different values of a. For example, when a equals 1, the quotient is 1; when a equals 2, the quotient is 2; and when a equals 3, the quotient is 3. Therefore, Tilde's claim that the quotient f prime of x divided by f of x equals 2 for the function f of x equals e to the power of 2x is indeed correct. Let's explore an alternative approach to verify Tilde's claim using logarithmic differentiation. This method is particularly useful for exponential and product functions. We start with our function f of x equals e to the power of 2x. Step 1: We take the natural logarithm of both sides. The natural logarithm of f of x equals the natural logarithm of e to the power of 2x, which simplifies to 2x because the natural logarithm and exponential function are inverse operations. Step 2: We differentiate both sides with respect to x. On the left side, we get the derivative of the natural logarithm of f of x, which equals f prime of x divided by f of x. On the right side, the derivative of 2x is simply 2. Therefore, f prime of x divided by f of x equals 2. Step 3: We can verify this result with our original calculation. We know that f prime of x equals 2e to the power of 2x, so the quotient f prime of x divided by f of x equals 2e to the power of 2x divided by e to the power of 2x, which simplifies to 2. This confirms that Tilde's claim is correct: the quotient f prime of x divided by f of x equals 2 for all values of x. Now, let's generalize our findings to other exponential functions. For any function of the form f of x equals e to the power of g of x, where g of x is any differentiable function, the derivative is f prime of x equals e to the power of g of x times g prime of x. When we calculate the quotient f prime of x divided by f of x, we get e to the power of g of x times g prime of x divided by e to the power of g of x, which simplifies to just g prime of x. This gives us a powerful pattern: the quotient f prime of x divided by f of x for an exponential function with base e equals the derivative of the exponent. Let's look at some examples. For f of x equals e to the power of 2x, the quotient equals 2, which is the derivative of 2x. For f of x equals e to the power of x squared, the quotient equals 2x, which is the derivative of x squared. For f of x equals e to the power of sine x, the quotient equals cosine x, which is the derivative of sine x. This pattern works for any exponential function with base e. Therefore, Tilde's claim that the quotient f prime of x divided by f of x equals 2 for the function f of x equals e to the power of 2x is correct, as it follows this general pattern. Let's summarize what we've learned about Tilde's claim. For the function f of x equals e to the power of 2x, we've shown that the quotient of the derivative f prime of x divided by the function f of x equals 2 for all values of x. This result follows from a general pattern: for any function of the form f of x equals e to the power of g of x, the quotient f prime of x divided by f of x equals g prime of x, which is the derivative of the exponent. We verified this result using three different approaches: direct differentiation, logarithmic differentiation, and by examining the general pattern for exponential functions. This property is particularly useful in various applications, such as solving differential equations and analyzing models of exponential growth or decay. In conclusion, Tilde's claim that the quotient f prime of x divided by f of x equals 2 for the function f of x equals e to the power of 2x is indeed correct.