The problem asks to find the derivative of the following two functions: a) $y = x \cdot \sin(2^{x+1})$ b) $y = x^2 \cdot 5^x \cdot \cos(e^{\sin(2^x)})$

1. Problem Description

The problem asks to find the derivative of the following two functions:

a) $y = x \cdot \sin(2^{x+1})$

b) $y = x^2 \cdot 5^x \cdot \cos(e^{\sin(2^x)})$

2. Solution Steps

a) $y = x \cdot \sin(2^{x+1})$

To find the derivative, we will use the product rule:

$(uv)' = u'v + uv'$, where $u = x$ and $v = \sin(2^{x+1})$.

$u' = (x)' = 1$

$v' = (\sin(2^{x+1}))' = \cos(2^{x+1}) \cdot (2^{x+1})'$

To find $(2^{x+1})'$, we use the formula $(a^x)' = a^x \ln(a)$.

$(2^{x+1})' = 2^{x+1} \cdot \ln(2) \cdot (x+1)' = 2^{x+1} \ln(2) \cdot 1 = 2^{x+1} \ln(2)$

So, $v' = \cos(2^{x+1}) \cdot 2^{x+1} \ln(2)$.

Now, we can use the product rule:

$y' = u'v + uv' = 1 \cdot \sin(2^{x+1}) + x \cdot \cos(2^{x+1}) \cdot 2^{x+1} \ln(2)$

$y' = \sin(2^{x+1}) + x \cdot 2^{x+1} \ln(2) \cdot \cos(2^{x+1})$

b) $y = x^2 \cdot 5^x \cdot \cos(e^{\sin(2^x)})$

We have a product of three functions, $u = x^2$, $v = 5^x$, and $w = \cos(e^{\sin(2^x)})$.

The derivative of a product of three functions is given by:

$(uvw)' = u'vw + uv'w + uvw'$.

$u' = (x^2)' = 2x$

$v' = (5^x)' = 5^x \ln(5)$

$w' = (\cos(e^{\sin(2^x)}))' = -\sin(e^{\sin(2^x)}) \cdot (e^{\sin(2^x)})'$

$(e^{\sin(2^x)})' = e^{\sin(2^x)} \cdot (\sin(2^x))'$

$(\sin(2^x))' = \cos(2^x) \cdot (2^x)'$

$(2^x)' = 2^x \ln(2)$

So, $w' = -\sin(e^{\sin(2^x)}) \cdot e^{\sin(2^x)} \cdot \cos(2^x) \cdot 2^x \ln(2)$

$y' = 2x \cdot 5^x \cdot \cos(e^{\sin(2^x)}) + x^2 \cdot 5^x \ln(5) \cdot \cos(e^{\sin(2^x)}) + x^2 \cdot 5^x \cdot (-\sin(e^{\sin(2^x)}) \cdot e^{\sin(2^x)} \cdot \cos(2^x) \cdot 2^x \ln(2))$

$y' = 2x \cdot 5^x \cdot \cos(e^{\sin(2^x)}) + x^2 \cdot 5^x \ln(5) \cdot \cos(e^{\sin(2^x)}) - x^2 \cdot 5^x \cdot \sin(e^{\sin(2^x)}) \cdot e^{\sin(2^x)} \cdot \cos(2^x) \cdot 2^x \ln(2)$

3. Final Answer

a) $y' = \sin(2^{x+1}) + x \cdot 2^{x+1} \ln(2) \cdot \cos(2^{x+1})$

b) $y' = 2x \cdot 5^x \cdot \cos(e^{\sin(2^x)}) + x^2 \cdot 5^x \ln(5) \cdot \cos(e^{\sin(2^x)}) - x^2 \cdot 5^x \cdot \sin(e^{\sin(2^x)}) \cdot e^{\sin(2^x)} \cdot \cos(2^x) \cdot 2^x \ln(2)$