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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Okay, Taye, let's break down each question separately. Question 1: Find from first principle the differential coefficient of the function y = 2x^2 with respect to x. Step 1: Recall the definition of the derivative from first principles. (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h) Given f(x) = 2x^2. First, find f(x+h) by substituting (x+h) into the function. f(x+h) = 2(x+h)^2 Expand the term (x+h)^2: (x+h)^2 = x^2 + 2xh + h^2 Now substitute this back into f(x+h): f(x+h) = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2 Step 2: Calculate the difference f(x+h) - f(x). f(x+h) - f(x) = (2x^2 + 4xh + 2h^2) - (2x^2) f(x+h) - f(x) = 4xh + 2h^2 Step 3: Divide the difference by h. (f(x+h) - f(x))/(h) = (4xh + 2h^2)/(h) Factor out h from the numerator: (h(4x + 2h))/(h) Cancel out h (since h ≠ 0 in the limit process): 4x + 2h Step 4: Take the limit as h 0. (dy)/(dx) = _h 0 (4x + 2h) Substitute h=0 into the expression: (dy)/(dx) = 4x + 2(0) (dy)/(dx) = 4x The differential coefficient is 4x. Question 2: Differentiate the function with respect to x, y = (1)/(x). Step 1: Rewrite the function using exponent notation. The term (1)/(x) can be written as x^-1. y = x^-1 Step 2: Apply the power rule for differentiation. The power rule states that if y = x^n, then (dy)/(dx) = nx^n-1. Here, n = -1. (dy)/(dx) = (-1) · x^-1-1 (dy)/(dx) = -1 · x^-2 Step 3: Rewrite the result with a positive exponent. The term x^-2 can be written as (1)/(x^2). (dy)/(dx) = -(1)/(x^2) The derivative is -(1)/(x^2). Question 3: Find the derivative of the function y = (3x+2) + e^2x. Step 1: Differentiate the first term, (3x+2), using the chain rule. The chain rule states that (d)/(dx)(f(g(x))) = f'(g(x)) · g'(x). For (3x+2), let f(u) = (u) and u = g(x) = 3x+2. The derivative of (u) with respect to u is (u). The derivative of 3x+2 with respect to x is 3. So, applying the chain rule: (d)/(dx)((3x+2)) = (3x+2) · (d)/(dx)(3x+2) (d)/(dx)((3x+2)) = (3x+2) · 3 (d)/(dx)((3x+2)) = 3(3x+2) Step 2: Differentiate the second term, e^2x, using the chain rule. For e^2x, let f(v) = e^v and v = g(x) = 2x. The derivative of e^v with respect to v is e^v. The derivative of 2x with respect to x is 2. So, applying the chain rule: (d)/(dx)(e^2x) = e^2x · (d)/(dx)(2x) (d)/(dx)(e^2x) = e^2x · 2 (d)/(dx)(e^2x) = 2e^2x Step 3: Combine the derivatives of the two terms. Since y = (3x+2) + e^2x, the derivative (dy)/(dx) is the sum of the derivatives of each term. (dy)/(dx) = (d)/(dx)((3x+2)) + (d)/(dx)(e^2x) (dy)/(dx) = 3(3x+2) + 2e^2x The derivative is 3(3x+2) + 2e^2x. That's 2 down. 3 left today — send the next one.