Fundamental Rules of Derivatives

The following are the fundamental rules of derivatives. Let us discuss them in detail.

Power Rule: The power rule of derivatives states that if a function is an algebraic expression raised to any power, say n, then the derivative has a power 1 less than the original function.  If y = xⁿ , where n > 0. Then dy/dx = n x ^n-1 . Example: x⁵ = 5x⁴

Sum Rule: The sum rule of derivatives states that if a derivative is the sum/difference of two functions, then it is equal to the sum/difference of its derivatives. dy/dx [u(x) ± v(x)]= du/dx ± dv/dx. 

Product Rule: The product rule of derivatives states that if a function is a product of two functions, then its derivative is the derivative of the second function multiplied by the first function added to the derivative of the first function multiplied by the second function. dy/dx [u(x) × v(x)] = u.dv/dx + v.du/dx. If y = x⁵ e^x , we have y’ = x⁵ e^x + e^x . 5x⁴ = e^x (x⁵ + 5x⁴)

Quotient Rule: The quotient rule of derivatives states that if a function is of the form u(x)/v(x), then the derivative is the difference between the second function × derivative of the first function and the first function × derivative of the second function divided by the square of the second function.  dy/dx [u(x) ÷ v(x)]= (v.du/dx- u.dv/dx)/ v² 

Constant multiple Rule: The constant multiple rule of derivatives states that if a function is the derivative multiplied by a constant then it is equal to the constant multiplied by its derivative d/dx [c(f(x)] = c. d/dx f(x), where c ≠ 0, and c is a constant. d/dx .5x² = 5. d/dx .x² = 5. 2x = 10 x.

Constant Rule: The constant rule of derivatives states that the derivative of any constant is 0. If y = k, where k is a constant, then dy/dx = 0. Suppose y = 4, y’ = 0