If $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(15)=2,\ f'(0)=3$, then $f'(5)$ will be

Question

Given that: \[ x = a \sin(2t) (1 + \cos(2t)), \quad y = a \cos(2t) (1 - \cos(2t)) \] Find $\frac{dy}{dx}$.

Answer 1

We are given that the function $f(x+y) = f(x)f(y)$ for all $x, y$. This is a functional equation that defines an exponential function.
We know $f(15) = 2$. Let's represent $f(x)$ in an exponential form as $f(x) = a^{x}$.
Given that $f(15) = a^{15} = 2$, we need to find the value of $a$.
Now, differentiate the function $f(x) = a^{x}$ with respect to $x$ to find $f'(x)$. Using the formula for differentiation of an exponential function, we get:
- $f'(x) = a^{x} \ln(a)$
We know $f'(0) = 3$. Thus, substituting $x = 0$ in the derivative:
- $f'(0) = a^{0} \ln(a) = \ln(a) = 3$
This gives us the equation:
- $\ln(a) = 3$
Exponentiate both sides to solve for $a$:
- $a = e^{3}$
Now, to find $f'(5)$, use:
- $f'(5) = a^{5} \ln(a) = (e^{3})^{5} \times 3 = e^{15} \times 3$
Given that $e^{15} = 2$ as derived from $f(15) = 2$, we have:
- $f'(5) = 2 \times 3 = 6$

Therefore, the value of $f'(5)$ is 6, which correctly matches the option $6$.

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