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Chapter 4: Q. 41 (page 399)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.

ddx鈭�18鈥�ln鈦�tdt

Short Answer

Expert verified

Ans: ddx鈭�18鈥�ln鈦�tdt=0

Step by step solution

01

Step 1. Given information.

given,

ddx鈭�18鈥�ln鈦�tdt

02

Step 2. The objective is to calculate the derivative. 

Now, if fis continuous on [a,b]then for allx鈭�[a,b],

ddx鈭�au(x)鈥�f(t)dt=f(u(x))u鈥�(x)

So,

role="math" f(u(t))=ln鈦�tf(u(8))=ln鈦�8u鈥�(x)=0f(u(x))u鈥�(x)=ln鈦�8(0)=0

03

Step 3. The derivative expression can be written as, 

ddx鈭�18鈥�ln鈦�tdt=0f(u(x))u鈥�(x)=0

Therefore, the value is 0

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