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.
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You're on a roll — here are the differentiations for each function: 1) y = x^2 x Step 1: Use the product rule, (d)/(dx)(uv) = u'v + uv'. Let u = x^2 and v = x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(x^2) = 2x. v' = (d)/(dx)( x) = - x. Step 3: Apply the product rule. (dy)/(dx) = (2x)( x) + (x^2)(- x) (dy)/(dx) = 2x x - x^2 x 2) y = e^x 3x Step 1: Use the product rule, (d)/(dx)(uv) = u'v + uv'. Let u = e^x and v = 3x. Step 2: Find the derivatives of u and v. u' = (d)/(dx)(e^x) = e^x. For v', use the chain rule: (d)/(dx)( 3x) = (3x) · (d)/(dx)(3x) = 3 3x. Step 3: Apply the product rule. (dy)/(dx) = (e^x)( 3x) + (e^x)(3 3x) (dy)/(dx) = e^x ( 3x + 3 3x) 3) y = e^3x nx Step 1: Use the product rule, (d)/(dx)(uv) = u'v + uv'. Let u = e^3x and v = nx. Step 2: Find the derivatives of u and v. For u', use the chain rule: (d)/(dx)(e^3x) = e^3x · (d)/(dx)(3x) = 3e^3x. For v', use the chain rule: (d)/(dx)( nx) = -(nx) · (d)/(dx)(nx) = -n nx. Step 3: Apply the product rule. (dy)/(dx) = (3e^3x)( nx) + (e^3x)(-n nx) (dy)/(dx) = e^3x (3 nx - n nx) 4) y = (x x)^2 Step 1: Use the chain rule. Let w = x x. Then y = w^2. (dy)/(dw) = 2w = 2x x. Step 2: Find (dw)/(dx) using the product rule for w = x x. Let u = x and v = x. u' = 1 and v' = x. (dw)/(dx) = (1)( x) + (x)( x) = x + x x. Step 3: Apply the chain rule (dy)/(dx) = (dy)/(dw) · (dw)/(dx). (dy)/(dx) = (2x x)( x + x x) (dy)/(dx) = 2x ^2 x + 2x^2 x x Using the identity (2x) = 2 x x: (dy)/(dx) = 2x ^2 x + x^2 (2x) 5) y = e^3x 4x Step 1: Use the product rule, (d)/(dx)(uv) = u'v + uv'. Let u = e^3x and v = 4x. Step 2: Find the derivatives of u and v. For u', use the chain rule: (d)/(dx)(e^3x) = e^3x · (d)/(dx)(3x) = 3e^3x. For v', use the chain rule: (d)/(dx)( 4x) = -(4x) · (d)/(dx)(4x) = -4 4x. Step 3: Apply the product rule. (dy)/(dx) = (3e^3x)( 4x) + (e^3x)(-4 4x) (dy)/(dx) = e^3x (3 4x - 4 4x) 6) y = x^3 (x^2+2) Step 1: Use the product rule, (d)/(dx)(uv) = u'v + uv'. Let u = x^3 and v = (x^2+2). Step 2: Find the derivatives of u and v. u' = (d)/(dx)(x^3) = 3x^2. For v', use the chain rule: (d)/(dx)((x^2+2)). Let w = x^2+2, so (dw)/(dx) = 2x. Then (d)/(dx)( w) = (1)/(w) · (dw)/(dx) = (1)/(x^2+2) · 2x = (2x)/(x^2+2). Step 3: Apply the product rule. (dy)/(dx) = (3x^2)((x^2+2)) + (x^3)((2x)/(x^2+2)) (dy)/(dx) = 3x^2 (x^2+2) + (2x^4)/(x^2+2) You can factor out x^2: (dy)/(dx) = x^2 (3 (x^2+2) + (2x^2)/(x^2+2)) What's next?