Determine the derivative of 2e^(x/2)*sin(4x)+π^3 using appropriate rules(no

Answered question

2022-02-20

Determine the derivative of 2e^(x/2)*sin(4x) using appropriate rules(no first principles).

Answer & Explanation

Vasquez

Vasquez

Expert2022-03-12Added 669 answers

Possible derivation:
ddx(2ex2sin(4x))

Factor out constants:
=2(ddx(ex2sin(4x)))

Use the product rule, ddx(uv)=vdudx+udvdx, where u=ex2 and v=sin(4x):
=2(ex2(ddx(sin(4x)))+(ddx(ex2))sin(4x))

Using the chain rule, ddx(sin(4x))=dsin(u)dududx, where u=4x and ddu(sin(u))=cos(u):
=2((ddx(ex2))sin(4x)+cos(4x)(ddx(4x))ex2)

Factor out constants:
=2((ddx(ex2))sin(4x)+4(ddx(x))ex2cos(4x))

The derivative of x is 1:
=2((ddx(ex2))sin(4x)+14ex2cos(4x))

Using the chain rule, ddx(ex2)=deudududx, where u=x2 and ddu(eu)=eu:
=2(4ex2cos(4x)+ex2(ddx(x2))sin(4x))

Factor out constants:
=2(4ex2cos(4x)+12(ddx(x))ex2sin(4x))

The derivative of x is 1:
=2(4ex2cos(4x)+112ex2sin(4x))

Simplify the expression:
Answer:  =ex2(8cos(4x)+sin(4x))

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?