This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Let's find the derivative of the given function f(x) = x^3(x^2) + (x)(e^x). We will use the sum rule, product rule, and chain rule for differentiation. The sum rule states that (d)/(dx)[g(x) + h(x)] = (d)/(dx)[g(x)] + (d)/(dx)[h(x)]. The product rule states that (d)/(dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x). The chain rule states that (d)/(dx)[f(g(x))] = f'(g(x))g'(x). Step 1: Differentiate the first term, x^3(x^2). Let u(x) = x^3 and v(x) = (x^2). Then u'(x) = (d)/(dx)[x^3] = 3x^2. To find v'(x), we apply the chain rule. Let w = x^2, so v(x) = (w). Then (dv)/(dw) = (w) and (dw)/(dx) = 2x. So, v'(x) = (d)/(dx)[(x^2)] = (x^2) · (2x) = 2x(x^2). Using the product rule for the first term: (d)/(dx)[x^3(x^2)] = u'(x)v(x) + u(x)v'(x) = (3x^2)(x^2) + (x^3)(2x(x^2)) = 3x^2(x^2) + 2x^4(x^2) Step 2: Differentiate the second term, (x)(e^x). Let p(x) = (x) and q(x) = (e^x). Then p'(x) = (d)/(dx)[(x)] = (1)/(x). To find q'(x), we apply the chain rule. Let z = e^x, so q(x) = (z). Then (dq)/(dz) = -(z) and (dz)/(dx) = e^x. So, q'(x) = (d)/(dx)[(e^x)] = -(e^x) · (e^x) = -e^x(e^x). Using the product rule for the second term: (d)/(dx)[(x)(e^x)] = p'(x)q(x) + p(x)q'(x) = ((1)/(x))(e^x) + ((x))(-e^x(e^x)) = ((e^x))/(x) - e^x(x)(e^x) Step 3: Combine the derivatives of both terms. f'(x) = (d)/(dx)[x^3(x^2)] + (d)/(dx)[(x)(e^x)] f'(x) = (3x^2(x^2) + 2x^4(x^2)) + (((e^x))/(x) - e^x(x)(e^x)) f'(x) = 3x^2(x^2) + 2x^4(x^2) + ((e^x))/(x) - e^x(x)(e^x) The final answer is f'(x) = 3x^2(x^2) + 2x^4(x^2) + ((e^x))/(x) - e^x(x)(e^x).