True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
$(a) T r u e o r F a l s e : \frac{d}{d x} (5) = 0 (b) T r u e o r F a l s e : \frac{d}{d r} (k s + r) = k (c) T r u e o r F a l s e : \frac{d}{d s} (k s + r) = k (d) T r u e o r F a l s e : \frac{d}{d x} {(3 x + 1)}^{k} = k {(3 x + 1)}^{k - 1} (e) T r u e o r F a l s e : \frac{d}{d x} (\frac{1}{x^{3}}) = \frac{1}{3 x^{2}} (f) T r u e o r F a l s e : I f f a n d g a r e d i f f e r e n t i a b l e f u n c t i o n s, t h e n (f (x) g (x))' = g' (x) f (x) + f' (x) g (x) (g) T r u e o r F a l s e : I f f a n d g a r e d i f f e r e n t i a b l e f u n c t i o n s, t h e n (\frac{g (x)}{h (x)})' = \frac{h (x) g' (x) - g (x) h' (x)}{(h {(x))}^{2}} (h) T r u e o r F a l s e : P r o v i n g t h e s u m r u l e f o r d i f f e r e n t i a t i o n i n v o l v e s t h e d e f i n i t i o n o f t h e d e r i v a t i v e, a l o t o f a l g e b r a c m a n i p u a t i o n s a n d t h e s u m r u l e f o r l i m i t s$

Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
$(a) T r u e o r F a l s e : \frac{d}{d x} (5) = 0 (b) T r u e o r F a l s e : \frac{d}{d r} (k s + r) = k (c) T r u e o r F a l s e : \frac{d}{d s} (k s + r) = k (d) T r u e o r F a l s e : \frac{d}{d x} {(3 x + 1)}^{k} = k {(3 x + 1)}^{k - 1} (e) T r u e o r F a l s e : \frac{d}{d x} (\frac{1}{x^{3}}) = \frac{1}{3 x^{2}} (f) T r u e o r F a l s e : I f f a n d g a r e d i f f e r e n t i a b l e f u n c t i o n s, t h e n (f (x) g (x))' = g' (x) f (x) + f' (x) g (x) (g) T r u e o r F a l s e : I f f a n d g a r e d i f f e r e n t i a b l e f u n c t i o n s, t h e n (\frac{g (x)}{h (x)})' = \frac{h (x) g' (x) - g (x) h' (x)}{(h {(x))}^{2}} (h) T r u e o r F a l s e : P r o v i n g t h e s u m r u l e f o r d i f f e r e n t i a t i o n i n v o l v e s t h e d e f i n i t i o n o f t h e d e r i v a t i v e, a l o t o f a l g e b r a c m a n i p u a t i o n s a n d t h e s u m r u l e f o r l i m i t s$

Accepted Answer

(a) The given statement is true as derivative of constant is 0

(b) The given statement is false because the derivative is 1

(c) The given statement is true because derivative is k

(d) The given statement is false because the derivative of 3x+1 is not taken .

(e) The derivative is not same. the given statement is false.

(f) The given statement is true because it represent the product rule

(g) The given statement is true because it represent the Quotient rule

(h) The given statement is true because the sum rule of limit is used for proving the sum rule of differentiation

Find Study Materials for

Create Study Materials

Select your language

Calculus

Answers without the blur.

Short Answer

Step by Step Solution

step 1:Given Information

Part (a): Step 1 : Derivative

Part(b): Step 1: Derivative

Part(c): Step 1: Derivative

Part(d) : Step 1: Derivative

Part(e): Step 1 : Derivative

Part f: step 1: Derivative

Part(g): Step 1 Derivative

Part(h): Step 1 : Derivative

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Calculus

Answers without the blur.

Short Answer

Step by Step Solution

step 1:Given Information

Part (a): Step 1 : Derivative

Part(b): Step 1: Derivative

Part(c): Step 1: Derivative

Part(d) : Step 1: Derivative

Part(e): Step 1 : Derivative

Part f: step 1: Derivative

Part(g): Step 1 Derivative

Part(h): Step 1 : Derivative

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Calculus

Pure Maths