If f(x) = x², then find f' from the first principle. 2. Complete the following statement. Derivative of sum is of derivative and derivative of difference is of derivative. 3. If y = xn, then find dy/dx. 4. If e-2x+y = 3xy, then find dy/dx.

Here are the solutions to the problems. 1. If f(x) = x^2, then find f' from the first principle. Step 1: Apply the definition of the derivative from the first principle. f'(x) = _h 0 (f(x+h) - f(x))/(h) Substitute f(x) = x^2 and f(x+h) = (x+h)^2. f' = _h 0 (x+h)^2 - x^2h Step 2: Expand (x+h)^2 and simplify the numerator. f' = _h 0 x^2 + 2xh + h^2 - x^2h f' = _h 0 2xh + h^2h Step 3: Factor out h from the numerator and cancel it with the denominator. f' = _h 0 (h(2x + h))/(h) f' = _h 0 2x+h Step 4: Evaluate the limit as h 0. f' = 2x + 0 f' = 2x 2. Complete the following statement. Derivative of sum is ______ of derivative and derivative of difference is ______ of derivative. Step 1: Recall the linearity properties of derivatives. The derivative of a sum of functions is the sum of their derivatives. The derivative of a difference of functions is the difference of their derivatives. Derivative of sum is sum of derivative and derivative of difference is difference of derivative. 3. If y = x^n, then find (dy)/(dx). Step 1: Apply the power rule for differentiation. Given y = x^n. (dy)/(dx) = nx^n-1 4. If e^-2x+y = 3xy, then find (dy)/(dx). Step 1: Differentiate both sides of the equation e^-2x+y = 3xy with respect to x implicitly. Using the chain rule for the left side and the product rule for the right side: (d)/(dx)(e^-2x+y) = e^-2x+y · (d)/(dx)(-2x+y) = e^-2x+y(-2 + (dy)/(dx)) (d)/(dx)(3xy) = (d)/(dx)(3x) · y + 3x · (d)/(dx)(y) = 3y + 3x(dy)/(dx) Equating the derivatives, the given line in the problem is: (d)/(dx)(-2x+y)e^-2x+y = 3xy' + 3y Step 2: Expand the left side and rearrange the equation to isolate terms with (dy)/(dx). -2e^-2x+y + e^-2x+y(dy)/(dx) = 3y + 3x(dy)/(dx) e^-2x+y(dy)/(dx) - 3x(dy)/(dx) = 3y + 2e^-2x+y Step 3: Factor out (dy)/(dx) and solve for it. (dy)/(dx)(e^-2x+y - 3x) = 3y + 2e^-2x+y (dy)/(dx) = 3y + 2e^-2x+ye^-2x+y - 3x Send me the next one 📸