Mathematical  Proof 


Derivative of Logarithmic Functions: Proving, mathematically, why the derivative in respect to x of (logarithm base "a" of x) is 1/xlna. 
Derivative of Logarithmic Functions (II): Mathematical proof of the derivative of logarithmic functions, starting from the the limit which defines a derivative. Also, explanation of why the "e" (Euler's number) is appears so often in calculus, showing that it appears naturally; the limit which defines "e". 



Derivative of e^x (I): Mathematical proof that the derivative of e^x is e^x, where "e" is the Euler's number (approximately 2.718... ). 
Derivative of e ^x (II) : Mathematically proving the derivative of e^x (where "e" is Euler's number), starting with the limit definition of a derivative. 


Polynomial Derivatives (I): Mathematical proof of the derivative of polynomial functions, with a restriction that "x" (from f(x)) must be greater than zero. 
Derivative of a^x: Proving mathematically that the derivative of the function a^x is (a^x)ln(a). 


Derivative of ln(x): Mathematically proving the derivative with respect to x of the function ln(x), where ln is the logarithm base "e" (the approximately 2.718... constant). 
The area of a circle: Mathematically proving the equation used to calculate the area of a circle. 


Euler's formula: Proving Euler's Formula (which relates trigonometry and complex numbers) using Taylor series expansions for some functions. 
Slopes of perpendicular lines: Coordinate geometry tells us that, if two lines are perpendicular one to another, their slopes are opposite reciprocals. 


Derivative of sine: Proving, using the limit definition of derivatives, that the derivative of sin(x) is cos(x). 
Derivative of Cosine: Proving, using the limit definition of derivatives, that the derivative of cos(x) is sin(x). 


Derivative of tangent: Proving, using the limit definition of derivatives, that the derivative of tan(x) is secē(x). 
Derivative of secant : Proving, using the limit definition of derivatives, that the derivative of sec(x) is tan(x)*sec(x). 


Derivative of cosecant : Proving, using the limit definition of derivatives, that the derivative of csc(x) is cotan(x)*csc(x). 
Derivative of cotangent : Proving, using the limit definition of derivatives, that the derivative of cotan(x) is cscē(x). 
