Derivative of function using evaluation theorem

[TsAmE]

Homework Statement

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function:

y = ∫ 0 to tanx ( ( t + (t)^1/2 )^1/2

Homework Equations

None

The Attempt at a Solution

I got my final answer to be:

y' = ( ( tanx + (tanx)^1/2 )^1/2

but the correct answer was:

y' = ( ( tanx + (tanx)^1/2 )^(1/2) * (secx)^2

 

[vela]

Hint: Chain rule.

 

[TsAmE]

My lecturer hasnt said anything about chain rule yet. Is there another normal way to do it? If not, could you please explain how to use the chain rule with it?

 

[vela]

Let F(t) is an antiderivative of $$\sqrt{t+\sqrt{t}}$$. Express

$$y=\int_0^{\tan x} \sqrt{t+\sqrt{t}}\,dt$$

in terms of F(t). Then differentiate it to find y'(x). When you take this derivative, you'll need to use the chain rule.

 

[TsAmE]

But if you differentiate that integral you will get $$\sqrt{t+\sqrt{t}}$$ before subing in 0 and tanx, I am not sure how you would do chain rule in reverse

 

[Mark44]

But if you differentiate that integral you will get $$\sqrt{t+\sqrt{t}}$$ before subing in 0 and tanx, I am not sure how you would do chain rule in reverse

If this were your equation, you would be correct.
$$y=\int_0^x \sqrt{t+\sqrt{t}} dt$$
For this equation
$$dy/dx = \sqrt{x+\sqrt{x}}$$

The trouble is, your integral is not from 0 to x, but is instead from 0 to tan(x). The idea with the fundamental theorem of calculus is that
$$d/dx \int_a^x f(t) dt = f(x)$$

What you have is
$$d/dx \int_a^{g(x)} f(t) dt = f(x)$$

They're not the same, and you need to use the chain rule for your problem. Note that your problem is NOT about finding the antiderivative of your integrand, and then substituting in tan(x) and 0.

 

[vela]

But if you differentiate that integral you will get $$\sqrt{t+\sqrt{t}}$$ before subing in 0 and tanx, I am not sure how you would do chain rule in reverse

The reason I suggested you write y(x) in terms of F(t) first and then differentiate it is to avoid the mistake you keep making. The fundamental theorem of calculus tells you

y(x) = F(tan x) - F(0)

Do you now see why you have to use the chain rule?