
Tangents, Derivatives and DifferentiationDerivation of The Basic Derivative RulesMost of the functions that we have to deal with when finding derivatives are combinations of the following elementary functions:
We now try to find the derivative of each of these elementary functions. The derivative of f(x) = c, where c is a constantLet f(x) = c, where c is a constant. Then Hence
The derivative of x^{n}From Example 2, the examples found in the "More" page following Example 2, and Exercise 1 you may have guessed that for any integer or fractional number n. In fact this is true for any real number n. Let us prove this.
where Q is a polynomial in x and h. Thus Hence for any nonnegative integer n This is known as the power rule and is in fact
true for any real number n. For example: (As proved in Exercise 1.) The derivative of sin xTo find the derivative of f(x) = sin x, we first need a couple of results:
The Newton's Quotient for the sin x is then From results (2) and (3) we see that this quotient tends to cos x as . Hence
The derivative of cos xTo find the derivative of f(x) = cosx, we
first need a couple of results:
From results (2) and (3) we see that this quotient tends to −sin x as . Hence
The derivative of e^{x}To find the derivative of f(x) = e^{x}, we recall the definition of Euler’s number e: where e is a nonfractional number roughly equal to 2.718281828459045235. That is Therefore, Hence
The derivative of ln xTo find the derivative of f(x) = ln x, we again recall the definition of Euler’s number e: where e is a nonfractional number roughly equal to 2.718281828459045235. Note that ln x is the inverse of the exponential function e^{x}. Thus ln e^{x} = x for all x. In particular,
Let f(x) = ln x. Then Let . Thus h = kx and the above becomes As h→0, k→0. Hence by the definition of e, from the above we have Therefore
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