Define y and y + y.

To find the derivative of y = 5x^3 - 8x using the first principle, we follow these steps: Step 1: Define y and y + y. We are given y = 5x^3 - 8x. Then, we replace x with x + x to find y + y: y + y = 5(x + x)^3 - 8(x + x) Step 2: Expand the expression for y + y. First, expand (x + x)^3: (x + x)^3 = x^3 + 3x^2( x) + 3x( x)^2 + ( x)^3 Substitute this back into the equation for y + y: y + y = 5(x^3 + 3x^2 x + 3x( x)^2 + ( x)^3) - 8(x + x) y + y = 5x^3 + 15x^2 x + 15x( x)^2 + 5( x)^3 - 8x - 8 x Step 3: Find y. Subtract y from y + y: y = (5x^3 + 15x^2 x + 15x( x)^2 + 5( x)^3 - 8x - 8 x) - (5x^3 - 8x) y = 5x^3 + 15x^2 x + 15x( x)^2 + 5( x)^3 - 8x - 8 x - 5x^3 + 8x Combine like terms: y = 15x^2 x + 15x( x)^2 + 5( x)^3 - 8 x Step 4: Divide y by x. Factor out x from the expression for y: ( y)/( x) = ( x (15x^2 + 15x x + 5( x)^2 - 8))/( x) Cancel x: ( y)/( x) = 15x^2 + 15x x + 5( x)^2 - 8 Step 5: Take the limit as x 0. The derivative (dy)/(dx) is defined as the limit of ( y)/( x) as x approaches 0: (dy)/(dx) = _ x 0 (15x^2 + 15x x + 5( x)^2 - 8) As x 0, the terms 15x x and 5( x)^2 become 0: (dy)/(dx) = 15x^2 + 15x(0) + 5(0)^2 - 8 (dy)/(dx) = 15x^2 - 8 The derivative of y = 5x^3 - 8x is: (dy)/(dx) = 15x^2 - 8 Send me the next one 📸