MATH 1225 Lecture Notes - Lecture 8: Differentiable Function, Constant Function, Power Rule

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28 May 2018

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Sec. 3.1 Derivatives of Polynomials and Exponential Functions

• Derivative of a constant function:

 

0c

EX:

 

• The Power rule: If

is any real number, then

 

1

nn nxx

EX:

 

4











4

• The Constant Multiple Rule: If

is a constant and

is a differentiable function, then

 

 

 

cxfc

d

EX:

 

d

 

7

 



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Document Summary

3. 1 derivatives of polynomials and exponential functions: derivative of a constant function: 5e d dx: the power rule: if n is any real number, then x n d dx nx. 4 x d dx: the constant multiple rule: if c is a constant and f is a differentiable function, then xfc c d dx. 25 d dx: the sum rule: if f and g are both differentiable, then xf xg d dx d dx xf. Xg d dx: the difference rule: if f and g are both differentiable, then xf xg d dx d dx xf. , then xf: definition of the number e: e is the number such that lim. 1: derivative of the natural exponential function: e x x e d dx. Is it the same as the domain of. Ex: find the derivative of f , the domain of f , and the domain of f xf. Ex: find a point on the graph of.

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10. Exactly how many of the following statements are always true? (i) If f is continuous at 0 ar :) dx > 0, then f is differentiable at 0. (ii) The definite integral of a continuous function h over [a, b] is a function H such that / h(x) dx = H(b) – Ha). (iii) A continuous function g has a local minimum at a number c if and only if c is a critical number of g. (iv) If a is a critical number of a polynomial, then the 2nd derivative test is always preferable over the 1st derivative test when checking for extrema at a. (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

MATH 1225 Lecture Notes - Lecture 8: Differentiable Function, Constant Function, Power Rule

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