If f(x) = x^sqrt(x^2+3)+10 , find the value of f'(1).

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justaguide | College Teacher | (Level 2) Distinguished Educator

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You can only ask one question at a time. I am providing the value of f'(1).

We have to find the derivative of f(x) = x^sqrt(x^2+3)+10

Let g(x) = x^sqrt(x^2+3)

ln (g(x)) = sqrt( x^2 + 3) ln x

1/(g(x)) * g'(x) = [sqrt( x^2 + 3)]'*ln x + sqrt( x^2 + 3)*[ln x]'

1/(g(x))*g'(x) = (1/2)*2x*(1/sqrt( x^2 + 3))*ln x + sqrt(x^2 + 3)*(1/x)

g'(x) = [(1/2)*2x*(1/sqrt( x^2 + 3))*ln x + sqrt(x^2 + 3)*(1/x)](x^sqrt(x^2+3))

=> g'(x) = [x^2*ln x +(x^2 + 3)](x^sqrt(x^2+3))/x*(sqrt( x^2 + 3))

=> g'(x) = [(x^sqrt(x^2+3))*x^2*ln x +(x^sqrt(x^2+3))(x^2 + 3)](x^sqrt(x^2+3))/x*(sqrt( x^2 + 3))

=> g'(x) = [(x^sqrt(x^2+3)+2)*ln x +(x^sqrt(x^2+3))(x^2 + 3)]/x*(sqrt( x^2 + 3))

We see that f'(x) = g'(x)  as 10 is a constant.

f'(x) = [(x^sqrt(x^2+3)+2)*ln x +(x^sqrt(x^2+3))(x^2 + 3)]/x*(sqrt( x^2 + 3))

f'(1) = 1*4/2

=> f'(1) = 2

The required value of f'(1) = 2.

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