Chapter 2: Q. 88 (page 199)

Prove the difference rule in two ways
a) using definition of the derivative
b) using sum and constant multiple rules

Short Answer

Expert verified

We prove the difference rules using the given rules

Step by step solution

Given information

We are given two function f anf g

Prove the difference rule using the definition of the derivative

We get,

$\lim_{h \to 0} \frac{(f - g) (x + h) - (f - g) (x)}{h} \lim_{h \to 0} \frac{f (x + h) - g (x + h) - f (x) + g (x)}{h} \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} - \lim_{h \to 0} \frac{g (x + h) - g (x)}{h} = f' (x) - g' (x) H e n c e p r o v e d$

Proof using sum and constant times rule

We have,

$W e a r e g i v e n (f - g)' (x)$

Which can also be written as

$(f + (- g))' (x) f' (x) + (- g' (x)) (S u m r u l e) f' (x) - g' (x) (c o n s \tan t r u l e)$

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Prove the difference rule in two ways
a) using definition of the derivative
b) using sum and constant multiple rules

Short Answer

Step by step solution

Given information

Prove the difference rule using the definition of the derivative

Proof using sum and constant times rule

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Prove the difference rule in two waysa) using definition of the derivativeb) using sum and constant multiple rules

Short Answer

Step by step solution

Given information

Prove the difference rule using the definition of the derivative

Proof using sum and constant times rule

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Prove the difference rule in two ways
a) using definition of the derivative
b) using sum and constant multiple rules