Chapter 7 Differential Calculus

Section 7.2 Standard Derivatives

7.2.2 Derivatives of Power Functions

In the last section, the derivative was introduced as the limit of the difference quotient. Accordingly, for a linear affine function (see Module 6, Section 6.2.4) f:, xf(x)=mx+b, where m and b are given numbers, we obtain for the derivative at the point x0 the value f'( x0 )=m. (Readers are invited to verify that fact themselves.)
For monomials xn with n1, it is easiest to determine the derivative using the difference quotient. Without any detailed calculation or any proof we state the following rules:
Derivative of xn 7.2.1
Let a natural number n and a real number r be given.
The constant function f: with xf(x):=r=r· x0 has the derivative f': with xf'(x)=0.
The function f: with xf(x):=r· xn has the derivative

f': with xf'(x)=r·n· xn-1 .

This differentiation rule is true for all n{0}.

Again, we leave the verification of these statements to the reader.
Example 7.2.2
Let us consider the function f: with xf(x)=5 x3 . According to the notation above, this is a function with r=5 and n=3. Thus for the value of the derivative at the point  x , we have

f'(x)=5·3 x3-1 =15 x2 .

For root functions, an equivalent statement holds. However, it should be noted that root functions are only differentiable for x>0 since the tangent line to the graph of the function at the point (0;0) is parallel to the y-axis and thus, it is not a graph of a function.
Derivative of x 1 n 7.2.3
For n with n0, the function f:[0;[ , xf(x):= x 1 n is differentiable for x>0, and we have

f': ]0;[ ,xf'(x)= 1 n · x 1 n -1 .

For n, root functions are described by f(x)= x 1 n . Of course, the differentiation rule given here also holds for n=1 or n=-1.
Example 7.2.4
The root function f:[0;[ with xf(x):=x= x 1 2 is differentiable for x>0. The value of the derivative at an arbitrary point x>0 is given by

f'(x)= 1 2 · x 1 2 -1 = 1 2 · x- 1 2 = 1 2·x .

The derivative at the point x0 =0 does not exist since the slope of the tangent line to the graph of f would be infinite there.

The tangent line to the graph of the given root function at the point (1;1) has the slope 1 21 = 1 2 .

For x>0, the statements above can be extended to exponents p with p0: The value f'(x) of the derivative of the function f with the mapping rule f(x)= xp is, for x>0,

f'(x)=p· xp-1 .