00:01
In this problem we are given that f is a differentiable function such that f of 0 is equal to 1 and f dash of 0 is equal to 4.
00:09
The first question is to determine h dash of 0 where h of x is defined as 2 raise to x times x square plus 1 times f of x.
00:23
So here we need to find the derivative h dash of x with respect to x.
00:28
For that we can consider h of x as the product of two functions, 2 raised to x times x square plus 1 and f of x.
00:37
And we can apply the product rule of differentiation to evaluate the derivative.
00:41
By product rule, the derivative d by d x of 2 functions u and v of x can be evaluated as the first function times the derivative of the second function plus the second function times the derivative of the first.
00:56
So here using the product rule we have this is the first function 2 raise to x times x square plus 1 times the derivative of the second function which is f dash of x plus the second function f of x times the derivative of the first so here again we need to apply product rule into this function so it is the first function 2 raise to x times the derivative of the second function that is 2x plus the second function x square plus 1 times the derivative of the first is the derivative of 2 raise to x and that is 2 raised to x log 2.
01:34
Now we need h dash of 0 so that is obtained by substituting x is equal to 0 into this function.
01:44
So it is 2 raise to 0 times 1 times f dash of 0 plus f of 0 times 2 raised to 0 times 2 raised to 0 times 0 plus 1 times 0 times 1 times...
