Rules of Differentiation

In order to simplify the task of finding derivatives, some general rules are developed to help finding derivatives with having to use the defination directly.

Derivatives of Polynomials

Derivative of Constant Function

Proof:
Derivative of Linear Function

Proof:
Derivative of Power Function

Proof when n is positive:

Proof when n is zero and x not equal to zero:

n can be any real number and proof will be included in the later part of rules.
Rule of Constant Multiple Function

Proof:
Rule of Sum of Functions

Proof:

The sum of functions can be extended to the sum of any functions by repeating the sum rule.
Rule of Difference of Functions

Proof if both functions are differentiable:

The difference of functions can be extended to the difference of any functions by repeating the difference rule. And the different rule can be obtained by applying the constant multiple rule to the sum rule.
Rule of Product of Functions

Proof if both functions are differentiable:

The proof is completed by subtracting and adding an addition expression in the numerator. Since both functions f and g are differentiable, they are continuous and f(x)=f(x+Δx) and g(x)=g(x+Δx) when Δx approaching zero. The product rule can also be extend to any number of functions.
Rule of Quotient of Functions

Proof if both functions are differentiable:

The proof is completed by subtracting and adding an addition expression in the numerator. Since function g is differentiable, it is continuous and g(x)=g(x+Δx) when Δx approaching zero.

Proof for the power function when n is negative:
Rule of Composite Function (The chain rule):

Proof if both functions are differentiable:

For g(x)

For f(g(x)):

Therefore:

Since function g is differentiable, it is continuous. When Δx approaching zero, εg equals zero and Δh is approaching zero. And therefore εf equals zero as Δx approaching zero also. The rule of composite function can be extended to any number of composite function and the derivative can be expressed in form of a chain.
Rule of Inverse Function

Proof if both functions are differentiable and x is the inverese function of y:

Since x=f-1(y) if and only if y=f(x):

Therefore:

Proof for the power function when n is rational number:

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Trigonometric

Derivatives of Trigonometric Functions

Trigonometric functions are very important in practical application. Trigonmetric function f is continuous when it is defined at every real numbers x in the domain. To determine the derivatives, the real number x is the angle of the trigonometric function and is measured in radian.

Limits and Trigonometric Functions

Trigonometric functions are functions of an angle. The three basic are sine cosine and tangent. The geometric definition of trigonometric functions can be related by sides of a triangle. One important limit related to finding derivatives of trigonometric functions is the ratio of value of a trigonometric function to the angle of the trigonometric function. Geometrically, the trigonometric functions can represented by constructing triangles in a unit circle with radius r=1.

Limits and Sine Function

According to the graph, consider the angle x in radian between 0 and π/2 when x approaching zero and not equal to zero, the relationship of the area of triangle w/sin x, the area of circular sector w/arc x, and the area of triangle w/tan x are:

As x decrease, cos x will increase. (sin x)/x will increase also but less than 1. Imply as x approach +0, limit of (sin x)/x exists. Rearrange the inequality:

As x approach +0, i.e. δ>x>0 imply:

Similarly as x approach -0, imply:

Therefore:

Limits and Cosine Function

Another important limit is:

Since cosine function and sine function are related by a right angle triangle and both functions are continuous in every numbe of the domain, the cosine function can be transformed through algebraic manipulations: