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Lesson Example Discussion Quiz: Class Homework

Step-3

Title: Differentiation

Grade: 1300-a Lesson: S2-L5

Explanation: Hello Students, time to practice and review the steps for the problem.

Lesson Steps

Step	Type	Explanation	Answer
1	Problem	Find the derivative of the function \$"h"("x") = (4"x"^3 + 2"x" - 1) / ("x"^2)\$.
2	Step	The given function	\$"h"("x") = (4"x"^3 + 2"x" - 1) / ("x"^2)\$
3	Step	To find the derivative of h(x)	\$"d"/"dx" "h"("x") = "d"/"dx" ((4"x"^3 + 2"x" - 1) / ("x"^2) ) \$
4	Formula:	The quotient rule states that if we have a function of the form \$"f"("x") = ("g"("x")) / ("h"("x"))\$, then the derivative f'(x) is given by:	\$ "f"'("x") = ("g"'("x") times "h"("x") - "g"("x") times "h"'("x"))/("h"("x"))^2 \$
5	Step	Now apply the quotient rule and differentiate the expression	\$"d"/"dx" "h"("x") = ("d"/"dx" (4"x"^3 + 2"x" - 1) ("x"^2) - (4"x"^3 + 2"x" - 1) "d"/"dx" ("x"^2))/("x"^2)^2 \$ \$"h"'("x") = (("d"/"dx" 4"x"^3 + "d"/"dx" 2"x" - "d"/"dx" 1) ("x"^2) - (4"x"^3 + 2"x" - 1) (2"x"))/("x"^4) \$
6	Step	After performing differentiation, simplify the resulting expression	\$ "h"'("x") = ((12"x"^2 + 2) ("x"^2) - (4"x"^3 + 2"x" - 1) (2"x"))/("x"^4) \$ \$"h"'("x") = (- 4"x"^4 - 2"x"^2 + 2"x")/("x"^4) \$
7	Step	The derivative of the function is	\$"h"'("x") = (- 4"x"^4 - 2"x"^2 + 2"x")/("x"^4) \$
8	Choice.A	This is wrong because it does not align with the accurate antiderivative for the provided function	\$ (8"x"^4 - 2"x"^3 + 4"x"^2 - "x") / "x"^4 \$
9	Choice.B	It seems that the accurate derivative for the provided function has been identified	\$ (-4"x"^4 - 2"x"^2 + 2"x") / "x"^4 \$
10	Choice.C	This option is incorrect as it diverges from the precise antiderivative corresponding to the given function	\$ 4 - 2/("x"^2)\$
11	Choice.D	This is wrong because it does not align with the accurate antiderivative for the provided function	\$ (4"x"^3 - "x"^2 + 2"x") / "x"^3 \$
12	Answer	Option	B
13	Sumup	Can you summarize what you’ve understood in the above steps?

Step

Type

Explanation

Answer

Problem

Find the derivative of the function \$"h"("x") = (4"x"^3 + 2"x" - 1) / ("x"^2)\$.

Step

The given function

\$"h"("x") = (4"x"^3 + 2"x" - 1) / ("x"^2)\$

Step

To find the derivative of h(x)

\$"d"/"dx" "h"("x") = "d"/"dx" ((4"x"^3 + 2"x" - 1) / ("x"^2) ) \$

Formula:

The quotient rule states that if we have a function of the form \$"f"("x") = ("g"("x")) / ("h"("x"))\$, then the derivative f'(x) is given by:

\$ "f"'("x") = ("g"'("x") times "h"("x") - "g"("x") times "h"'("x"))/("h"("x"))^2 \$

Step

Now apply the quotient rule and differentiate the expression

\$"d"/"dx" "h"("x") = ("d"/"dx" (4"x"^3 + 2"x" - 1) ("x"^2) - (4"x"^3 + 2"x" - 1) "d"/"dx" ("x"^2))/("x"^2)^2 \$

\$"h"'("x") = (("d"/"dx" 4"x"^3 + "d"/"dx" 2"x" - "d"/"dx" 1) ("x"^2) - (4"x"^3 + 2"x" - 1) (2"x"))/("x"^4) \$

Step

After performing differentiation, simplify the resulting expression

\$ "h"'("x") = ((12"x"^2 + 2) ("x"^2) - (4"x"^3 + 2"x" - 1) (2"x"))/("x"^4) \$

\$"h"'("x") = (- 4"x"^4 - 2"x"^2 + 2"x")/("x"^4) \$

Step

The derivative of the function is

\$"h"'("x") = (- 4"x"^4 - 2"x"^2 + 2"x")/("x"^4) \$

Choice.A

This is wrong because it does not align with the accurate antiderivative for the provided function

\$ (8"x"^4 - 2"x"^3 + 4"x"^2 - "x") / "x"^4 \$

Choice.B

It seems that the accurate derivative for the provided function has been identified

\$ (-4"x"^4 - 2"x"^2 + 2"x") / "x"^4 \$

Choice.C

This option is incorrect as it diverges from the precise antiderivative corresponding to the given function

\$ 4 - 2/("x"^2)\$

Choice.D

This is wrong because it does not align with the accurate antiderivative for the provided function

\$ (4"x"^3 - "x"^2 + 2"x") / "x"^3 \$

Answer

Option

Sumup

Can you summarize what you’ve understood in the above steps?