Derivative computations

The formula lim f(x)-f(x+h)/h when  x→h that defines the derivative of a function f implies tedious calculations to calculate the derivative of some types of functions and combinations of functions..

Therefore some formulas have been established to determine the derivatives of a combination of functions and some specific types of functions.

The formulas for the constant function and the power functions are called respectively constant rule and power rule. The formulas for the sum, product and quotient of functions are called respectively addition rule, product rule and quotient rule. The derivative of a composition of 2 functions f and g is called the chain rule. It is an extension of the power rule The trigonometric, logarithmic and exponential functions have their specific formula.

The derivative of an implicit function is called implicit differentiation.

It is essential to memorize the formulas. Otherwise, it would be difficult to calculate the derivatives of these particular functions. Today we are going to limiting ourselves to the learning of the basic formulas: constant, power, sum, product and quotient rule.

Derivative of a constant

The derivative of the function constant is 0. If f(x) = c the derivative of f(x) is 0. We write:  f′(x) = 0.

The Power rule

The derivative of the function power defined by f(x) =  xⁿ is equal to n multiplied by x to the power of n-1. The formula is .f’(x) = nx^n-1

Derivative of the product of a constant by a function

The derivative of the product of a function by a constant is equal to the product of the constant by the derivative of the function power.

If  f(x) = axⁿ its derivative is f’(x) = ax^n-1


Derivative of the function f(x) = x

The derivative of the function f(x) = x can be calculated using the formula for the derivative of the function power. In order to use this formula we have to write f(x) = x as the function power. We write f(x) = x as f(x) = x¹ 
By applying the formula for the function power we obtain f’(x) = x^1-1 
 f’(x) = x⁰ ⇒ f’(x) = 1 

Derivative of a sum of functions

If f. g. h,;;; are differentiable for any value of x of their domain the derivative of the sum of these functions is f’+g’+h’+ ....

Derivative of the product of 2 functions

If f and g are differentiable for any value x of their domain the derivative of the product f.g is fg’+gf’

Derivative of the quotient of 2 functions

If f and g are differentiable for any value of their domain the derivative of the quotient f/g is (f∕g)’ = f’g-gf’∕g²

These formulas have to be demonstrated and the learners have to do some exercises to apply them. If anyone is interested in learning more subscribe to these courses via this link Free Introductory Calculus Course and Complete Calculus Course

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