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
To differentiate the expression x - x^3 x with respect to x, we will apply the rules of differentiation, including the product rule for the second term. Let y = x - x^3 x. Step 1: Differentiate the first term, x. The derivative of x with respect to x is x. (d)/(dx)( x) = x Step 2: Differentiate the second term, x^3 x, using the product rule. The product rule states that if u(x) = f(x)g(x), then (du)/(dx) = f'(x)g(x) + f(x)g'(x). Let f(x) = x^3 and g(x) = x. Then, f'(x) = 3x^2 and g'(x) = - x. Applying the product rule: (d)/(dx)(x^3 x) = (3x^2)( x) + (x^3)(- x) (d)/(dx)(x^3 x) = 3x^2 x - x^3 x Step 3: Combine the derivatives of both terms. (dy)/(dx) = (d)/(dx)( x) - (d)/(dx)(x^3 x) (dy)/(dx) = x - (3x^2 x - x^3 x) (dy)/(dx) = x - 3x^2 x + x^3 x Step 4: Factor out x from the terms containing it. (dy)/(dx) = (1 - 3x^2) x + x^3 x Comparing this accurate result with the given options: a. (1-3x) x + x^2 x b. (1-3x) x + x^3 x c. (1+3x) x + x^3 x d. (1-3x) x - x^3 x e. (1-2x) x + x^3 x The mathematically correct derivative is (1 - 3x^2) x + x^3 x. None of the provided options exactly match this result. Option b, (1-3x) x + x^3 x, is the closest, differing only in the power of x in the coefficient of x (it has 3x instead of 3x^2). (1 - 3x^2) x + x^3 x That's 2 down. 3 left today — send the next one.