If f(x) = x^2 cos x + sin x, find f”(x). f”(x) = 4 cos x -...

If $f(x) = x^2 \cos x + \sin x$, find $f''(x)$. $f''(x) = 4 \cos x - 4x \sin x + x^2 \cos x - \sin x$ $f''(x) = 2 \cos x - 4x \sin x + x^2 \cos x - \sin x$ $f''(x) = 2 \cos x + 2x \sin x - x^2 \cos x - \sin x$ $f''(x) = 2 \cos x - 4x \sin x - x^2 \cos x - \sin x$ $f''(x) = 4 \cos x + 4x \sin x - 2x^2 \cos x - \sin x$

Transcript

00:01 In this problem of function f x is given and we have to find the values for the constant in derivative.

00:07 So for the first derivative f dash x, we differentiate the function f x is equal to 2 multiplied by sign of 3x minus 4 multiplied by cos of 3x and we differentiate this with respect to x.

00:28 Then we get f dash of x is equal to 6 multiplied by cause of 3x.

00:33 Plus 12 multiplied by sine of 3x and then we compare it to the equation given as a problem that is a dash of x is equal to a sine b x plus c cos d x and on comparing these two equations you will get the values of a, b, c and b.

00:56 So a is equal to the coefficient of sine term that is 12 b is equal to sine of 3x it is given so b is equal to 3 then c is equal to coefficient coefficient of constant that is 6 and d is equal to 3.

01:11 So these are the values of constants for the first derivative of fx.

01:17 Then we move to the second derivative.

01:19 Here we are going to differentiate f dash of x with respect to x and we will get f double dash of x is equal to minus 18 multiplied by sine of 3x plus 36 multiplied by cos of 3x and we compare this to the equation given in the problem that is f double dash of x.

01:39 Is equal to f of sine of g x plus h multiplied by cos of j x and on comparing these two equations will get the value of f g h and j so f is equal to coefficient of sine term that is minus 18 g is equal to three h is equal to coefficient of cost term that is equal to 36 and j is equal to three so these are the values of constants for the second derivative of f x then we move to the third derivative and differentiate f double dash x with respect to x so we will get f3x is equal to minus 50 4 multiplied by cos of 3x minus 108 multiplied by sine of 3x and we compare this with the equation given in the problem that is f3 x is equal to k multiplied by sine of lx plus m multiplied by x x x plus m multiplied by cos of nx.

02:48 And on comparing these two equations, we can get the value of klm and n...

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