1

Problem

Find the derivative of the function \$h(x) = e^(2x) \times sin(x)\$.

2

Step

The given function

\$h(x) = e^(2x) \times sin(x)\$

3

Step

To find the derivative of h(x)

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

4

Step

The product rule states that if we have a function of the form \$h(x) = f(x) \times 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^(2x), g(x) = sinx \$

6

Step

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

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

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

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

7

Solution

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

8

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

9

Choice-A

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

Wrong \$ e^(2x) \times sin(x) + e^(2x) \times cos(x) \$

10

Choice-B

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

Wrong \$ 2e^(2x) \times sin(x) - e^(2x) \times cos(x) \$

11

Choice-C

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

Correct \$ 2e^(2x) \times sin(x) + e^(2x) \times cos(x) \$

12

Choice-D

This option matches the solution we obtained using the product rule

Wrong \$ 2xe^(2x) \times sin(x) + e^(2x) \times cos(x) \$

13

Answer

Option

C

14

Sumup

Please summarize choices