Differentiation in Basic Calculus

Hi my name is Loverne Grace Mae S. Mopal from Grade 12 STEM, St. Hubertus. And in this website I will give you a further explanation about Differentiation.

• Meaning
• Rules
• Real Life Application

In calculus, differentiation is one of the two important concepts apart from integration. Differentiation is a method of finding the derivative of a function. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on one of its variables. The most common example is the rate change of displacement with respect to time, called velocity. The opposite of finding a derivative is anti-differentiation.If x is a variable and y is another variable, then the rate of change of x with respect to y is given by dy/dx. This is the general expression of derivative of a function and is represented as f'(x) = dy/dx, where y = f(x) is any function.

RULE #1 : CONSTANT RULEConstant Rule states that if f(x) = c is a constant, then f’(x) = 0

RULE #2 : POWER RULEThe rule says that for any term of the form xⁿ, the derivative of the term is (n)x ^(n-1)

RULE #3 : CONSTANT MULTIPLEThis rule says the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
f’(x) = (a • n)x^(n-1)

RULE #4 : SUM & DIFFERENCEThe derivative of the sum and difference of a differentiable function is the sum and difference of the derivatives of the function.
f’(x) = g’(x) + h’(x) => sum
f’(x) = g’(x) - h’(x) => difference

RULE #5 : PRODUCT RULELet f(x) and g(x) be two functions and h be small increments in the function we get f(x + h) and g(x + h).F'(x) = f(x)g' (x) + g(x)f '(x)

RULE #6 : QUOTIENT RULEA Quotient Rule is stated as the ratio of the quantity of the denominator times the derivative of the numerator function minus the numerator times the derivative of the denominator function to the square of the denominator function.[f(x) • g’(x)] - [f’(x) • g(x)] / [g(x)]^2

RULE #7 : TRIGONOMETRIC FUNCTIONSThe differentiation of trigonometric functions can be done using the derivatives of sin x and cos x by applying the quotient rule. The differentiation formulas of the six trigonometric functions are listed in the video.

RULE #8 : EXPONENTIAL FUNCTIONSThe derivative of exponential function f(x) = a^{x a > 0 is the product of exponential function a}x and natural log of a, that is, f'(x) = a^{x ln a.}Mathematically, the derivative of exponential function is written as d(ax)/dx = (ax)' = a^x ln a.