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Lesson Example Discussion Quiz: Class Homework

Step-4

Title: Differentiation

Grade: 10-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") = "e"^(2"x") * "sin"("x")\$.
2	Step	The given function	\$"h"("x") = "e"^(2"x") * "sin"("x")\$
3	Step	To find the derivative of h(x)	\$"d"/"dx" "h"("x") = "d"/"dx" ("e"^(2"x") times "sin"("x")) \$
4	Formula:	The product rule states that if we have a function of the form \$"h"("x") = "f"("x") * "g"("x")\$, the a derivative is given by	\$"h"'("x") = "f"'("x") times "g"("x") + "f"("x") times "g"'("x") \$
5	Step	Now compare	\$"f"("x") = "e"^(2"x") , "g"("x") = "sinx" \$
6	Step	Let’s apply the product rule to h(x) then after differentiation	\$ "h"'("x") = "d"/"dx" ("e"^(2"x")) * "sin"("x") + "e"^(2"x") * "d"/"dx" "sinx" \$ \$ "h"'("x") = ("e"^(2"x") * 2) * "sin"("x") + "e"^(2"x") * "cos"("x") \$ \$ "h"'("x") = 2"e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$
7	Step	Therefore, the derivative of h(x) is \$ "h"'("x") = 2"e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$.
8	Choice.A	This choice is incorrect because the first term is missing a factor of 2 multiplying the expression \$2"e"^(2"x")"sin"("x")\$	\$ "e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$
9	Choice.B	This option has a factor of 2 for the first term but incorrectly uses subtraction instead of addition for the second term	\$ 2"e"^(2"x") * "sin"("x") - "e"^(2"x") * "cos"("x") \$
10	Choice.C	This option matches the solution we obtained using the product rule	\$ 2"e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$
11	Choice.D	This option introduces an unnecessary factor of x multiplying the first term \$ (2"xe"^(2"x") "sin"("x"))\$ which wasn’t present in the original expression or the product rule derivation	\$ 2"xe"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$
12	Answer	Option	C
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") = "e"^(2"x") * "sin"("x")\$.

Step

The given function

\$"h"("x") = "e"^(2"x") * "sin"("x")\$

Step

To find the derivative of h(x)

\$"d"/"dx" "h"("x") = "d"/"dx" ("e"^(2"x") times "sin"("x")) \$

Formula:

The product rule states that if we have a function of the form \$"h"("x") = "f"("x") * "g"("x")\$, the a derivative is given by

\$"h"'("x") = "f"'("x") times "g"("x") + "f"("x") times "g"'("x") \$

Step

Now compare

\$"f"("x") = "e"^(2"x") , "g"("x") = "sinx" \$

Step

Let’s apply the product rule to h(x) then after differentiation

\$ "h"'("x") = "d"/"dx" ("e"^(2"x")) * "sin"("x") + "e"^(2"x") * "d"/"dx" "sinx" \$

\$ "h"'("x") = ("e"^(2"x") * 2) * "sin"("x") + "e"^(2"x") * "cos"("x") \$

\$ "h"'("x") = 2"e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$

Step

Therefore, the derivative of h(x) is \$ "h"'("x") = 2"e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$.

Choice.A

This choice is incorrect because the first term is missing a factor of 2 multiplying the expression \$2"e"^(2"x")"sin"("x")\$

\$ "e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$

Choice.B

This option has a factor of 2 for the first term but incorrectly uses subtraction instead of addition for the second term

\$ 2"e"^(2"x") * "sin"("x") - "e"^(2"x") * "cos"("x") \$

Choice.C

This option matches the solution we obtained using the product rule

\$ 2"e"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$

Choice.D

This option introduces an unnecessary factor of x multiplying the first term \$ (2"xe"^(2"x") "sin"("x"))\$ which wasn’t present in the original expression or the product rule derivation

\$ 2"xe"^(2"x") * "sin"("x") + "e"^(2"x") * "cos"("x") \$

Answer

Option

Sumup

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