Find the derivative using the derivative rules ?

enter image source here

1 Answer
Darshan Senthil
Dec 8, 2017

10xsin−1(4x)+20x2√1−16x2

Explanation:

There's 3 rules you need to use here:

  1. Product Rule
  2. Power Rule
  3. Chain Rule

When taking derivatives, you always want to work outside-in. The outermost rule here is the product rule , so you'd use that.

Video, in case you need it:

The general rule is:

ddx[f(x)⋅g(x)]=ddx[f(x)]⋅g(x)+ddx[g(x)]⋅f(x)

So:
ddx(5x2sin−1(4x))=ddx(5x2)⋅sin−1(4x)+ddxsin−1(4x)⋅5x2

That gives you two derivatives you need to evaluate.

To evaluate the first of these two derivatives, you'll need to use the power rule.

Video, in case you need it:

The general rule is:

ddxxa=axa−1

So, what you'd have is:

ddx5x2=2(5x2−1)=10x

To evaluate the second of these two derivatives, you'll need to employ the chain rule.

Video, in case you need it:

The general rule is:

ddxf(g(x))=f'(g(x))⋅g'(x)

So:

ddxsin−1(4x)=ddx(sin−1(4x))⋅ddx(4x)

1√1−(4x)2⋅4

=4√1−16x2

Now, we just put it all together:

ddx(5x2sin−1(4x))=10x⋅sin−1(4x)+4√1−16x2⋅5x2

Simplify, and it all boils down to:

=10xsin−1(4x)+20x2√1−16x2

Hope that helps :)

Related questions
Impact of this question
2478 views around the world
You can reuse this answer
Creative Commons License