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 y = ax^n using the first principle method, we use the definition of the derivative. Step 1: State the definition of the derivative from first principles. The derivative (dy)/(dx) is defined as: (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x) = ax^n and f(x+h) = a(x+h)^n into the formula. (dy)/(dx) = _h 0 (a(x+h)^n - ax^n)/(h) Step 3: Expand (x+h)^n using the binomial theorem and simplify the numerator. The binomial expansion of (x+h)^n is x^n + nx^n-1h + (n(n-1))/(2!)x^n-2h^2 + + h^n. (dy)/(dx) = _h 0 a ( x^n + nx^n-1h + (n(n-1))/(2!)x^n-2h^2 + + h^n ) - ax^nh (dy)/(dx) = _h 0 ax^n + anx^n-1h + a(n(n-1))/(2!)x^n-2h^2 + + ah^n - ax^nh The ax^n terms cancel out: (dy)/(dx) = _h 0 anx^n-1h + a(n(n-1))/(2!)x^n-2h^2 + + ah^nh Step 4: Divide by h and evaluate the limit as h 0. Divide each term in the numerator by h: (dy)/(dx) = _h 0 ( anx^n-1 + a(n(n-1))/(2!)x^n-2h + + ah^n-1 ) As h 0, all terms containing h will become zero, except for the first term anx^n-1. (dy)/(dx) = anx^n-1 The derivative of y = ax^n using the first principle method is: (dy)/(dx) = anx^n-1 What's next? 📸