Identify Function: We need to find the derivative of the function f(x) = x*2^(x) with respect to x. This requires the use of the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Apply Product Rule: Let's denote the first function as u(x) = x and the second function as v(x) = 2^(x). According to the product rule, the derivative of f(x) = u(x)*v(x) is given by f'(x) = u'(x)*v(x) + u(x)*v'(x).Find Derivative of u(x): First, we find the derivative of u(x) = x with respect to x. The derivative of x with respect to x is 1, so u'(x) = 1.Find Derivative of v(x): Next, we find the derivative of v(x) = 2^(x) with respect to x. The derivative of an exponential function a^(x) with base a is a^(x)*ln(a), where ln denotes the natural logarithm. Therefore, v'(x) = 2^(x)*ln(2).Apply Product Rule Again: Now we apply the product rule: f'(x) = u'(x)*v(x) + u(x)*v'(x). Substituting the derivatives we found, we get f'(x) = 1*2^(x) + x*(2^(x)*ln(2)).Simplify Expression: Simplify the expression: f'(x) = 2^(x) + x*2^(x)*ln(2).Final Answer: The final answer is the simplified derivative of the function f(x) = x*2^(x), which is f'(x) = 2^(x) + x*2^(x)*ln(2).

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$\frac{d}{dx}(x \cdot 2^{x})$

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Algebra 1

Multiplication with rational exponents

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\frac{d}{dx}(x \cdot 2^{x})

Identify Function: We need to find the derivative of the function $f(x) = x \cdot 2^{x}$ with respect to $x$ . This requires the use of the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Apply Product Rule: Let's denote the first function as $u(x) = x$ and the second function as $v(x) = 2^{x}$ . According to the product rule, the derivative of $f(x) = u(x)\cdot v(x)$ is given by $f'(x) = u'(x)\cdot v(x) + u(x)\cdot v'(x)$ .
Find Derivative of $u(x)$ : First, we find the derivative of $u(x) = x$ with respect to $x$ . The derivative of $x$ with respect to $x$ is $1$ , so $u'(x) = 1$ .
Find Derivative of $v(x)$ : Next, we find the derivative of $v(x) = 2^{x}$ with respect to $x$ . The derivative of an exponential function $a^{x}$ with base $a$ is $a^{x}\cdot\ln(a)$ , where $ln$ denotes the natural logarithm. Therefore, $v'(x) = 2^{x}\cdot\ln(2)$ .
Apply Product Rule Again: Now we apply the product rule: $f'(x) = u'(x)\cdot v(x) + u(x)\cdot v'(x)$ . Substituting the derivatives we found, we get $f'(x) = 1\cdot 2^{x} + x\cdot (2^{x}\cdot \ln(2))$ .
Simplify Expression: Simplify the expression: $f'(x) = 2^{x} + x\cdot2^{x}\cdot\ln(2)$ .
Final Answer: The final answer is the simplified derivative of the function $f(x) = x\cdot2^{x}$ , which is $f'(x) = 2^{x} + x\cdot2^{x}\cdot\ln(2)$ .