Differentiation

Answer

Use f'(x) = lim(h→0) [(f(x+h) - f(x))/h]. Step 1: f(x+h) = (x+h)^2 = x^2 + 2xh + h^2. Step 2: f(x+h) - f(x) = 2xh + h^2. Step 3: (f(x+h) - f(x))/h = 2x + h. Step 4: Take the limit as h → 0. Answer: f'…

Answer

Use f'(x) = lim(h→0) [(f(x+h) - f(x))/h]. Step 1: f(x+h) = 1/(x+h). Step 2: f(x+h) - f(x) = 1/(x+h) - 1/x = (x - (x+h))/(x(x+h)) = -h/[x(x+h)]. Step 3: Divide by h: [-h/(x(x+h))]/h = -1/[x(x+h)]. Step…

Answer

Use d/dx(x^n) = nx^(n-1). Step 1: Here n = 7. Step 2: Multiply by the exponent and reduce the power by 1. Answer: d/dx(x^7) = 7x^6. Quick check: the power drops from 7 to 6, which is correct.

Answer

Use the exponential derivative rule. Step 1: d/dx(e^x) = e^x. Step 2: The first-principle proof uses the limit lim(h→0) (e^h - 1)/h = 1. Step 3: So the derivative of e^x remains e^x. Answer: d/dx(e^x)…

Answer

Use d/dx(a^x) = a^x log a. Step 1: Identify the base as a constant. Step 2: Multiply the original function by log a. Example: d/dx(3^x) = 3^x log 3. Answer: d/dx(a^x) = a^x log a.

Answer

Use d/dx(log_e x) = 1/x. Step 1: The derivative is defined for x > 0. Step 2: Differentiate. Example: d/dx(log_e x) = 1/x. Answer: d/dx(log_e x) = 1/x, for x > 0.

Answer

Use the product rule: d/dx(F1F2) = F1(dF2/dx) + F2(dF1/dx). Step 1: Let F1 = x^2 + 1 and F2 = x^3 - 4. Step 2: dF1/dx = 2x, dF2/dx = 3x^2. Step 3: Apply the rule. Step 4: f'(x) = (x^2 + 1)(3x^2) + (x^…

Answer

Use the quotient rule: d/dx(F2/F1) = [F1(dF2/dx) - F2(dF1/dx)] / F1^2. Step 1: Let F2 = x^2 + 1 and F1 = x - 2. Step 2: dF2/dx = 2x, dF1/dx = 1. Step 3: Apply the rule. Step 4: f'(x) = [(x - 2)(2x) - …