Let f(x)=sqrt(3+2x)
f′(5)=
f(x) = (3+2x)^(½)
f'(x) = ½(3+2x)^(-½) * 2 = (3+2x)^(-½)
f'(5) = (3+10)^(-½) = 1/√13
To find the derivative of a function, we can use the power rule for differentiation. The power rule states that if we have a function of the form f(x) = ax^n, where a and n are constants, then the derivative is given by f'(x) = nax^(n-1).
In this case, our function is f(x) = sqrt(3+2x). To find the derivative f'(x), we need to apply the power rule.
First, rewrite the function as f(x) = (3+2x)^(1/2). Now we can see that the function has a fractional exponent of 1/2.
To differentiate this, we multiply the exponent by the coefficient of x and then subtract 1 from the exponent:
f'(x) = (1/2)(3+2x)^(1/2 - 1) * (2x) = (1/2)(3+2x)^(-1/2) * 2x
To find f'(5), we substitute x = 5 into the expression for f'(x):
f'(5) = (1/2)(3+2(5))^(-1/2) * 2(5) = (1/2)(3+10)^(-1/2) * 10
Now we can simplify:
f'(5) = (1/2)(13)^(-1/2) * 10
To evaluate (13)^(-1/2), we can rewrite it as 1/(sqrt(13)):
f'(5) = (1/2)(1/sqrt(13)) * 10
Simplifying further:
f'(5) = 10/(2*sqrt(13))
f'(5) = 5/sqrt(13)
Therefore, f'(5) is equal to 5/sqrt(13).