find h'(2) given that f(2) = -3, g(2) = 4, f'(2)= -2 and g'(2)= 7.
A. h(x) = 5f(x) - 4g(x)
B. h(x) = f(x)g(x)
C. h(x) = f(x)/g(x)
D. h(x) = g(x)/ 1+ f(x)
simply chart it out.
f(x) g(x) f'(x) g'(x)
2| -3 4 -2 7
then plug them into the equations.
A. (5)-3 - 4(4)
-15 - 16 = -31
B. (-3)4 = -12
C. -3/4
D. 4/1-3
= 4/-2 = -2
well the chart didn show up right but when x = 2 then
f(x) = -3
g(x) = 4
f'(x) = -2
g'(x) = 7
then plug them into the equations.
A. (5)-3 - 4(4)
-15 - 16 = -31
B. (-3)4 = -12
C. -3/4
D. 4/1-3
= 4/-2 = -2
To find h'(2), we need to find the derivative of the function h(x) and then evaluate it at x = 2.
Let's find the derivative of each option:
A. h(x) = 5f(x) - 4g(x)
To find h'(x), we can use the linearity of the derivative, which states that if f(x) and g(x) are differentiable functions and c and d are constants, then the derivative of c*f(x) - d*g(x) is c*f'(x) - d*g'(x).
So, differentiating h(x) = 5f(x) - 4g(x), we get h'(x) = 5f'(x) - 4g'(x)
Substituting the given values, h'(2) = 5*f'(2) - 4*g'(2) = 5*(-2) - 4*7 = -10 - 28 = -38
B. h(x) = f(x)g(x)
To find h'(x), we need to use the product rule, which states that if f(x) and g(x) are differentiable functions, then the derivative of f(x)*g(x) is f'(x)*g(x) + f(x)*g'(x).
So, differentiating h(x) = f(x)g(x), we get h'(x) = f'(x)*g(x) + f(x)*g'(x)
Substituting the given values, h'(2) = f'(2)*g(2) + f(2)*g'(2) = -2*4 + (-3)*7 = -8 - 21 = -29
C. h(x) = f(x)/g(x)
To find h'(x), we need to use the quotient rule, which states that if f(x) and g(x) are differentiable functions, then the derivative of f(x)/g(x) is (f'(x)*g(x) - f(x)*g'(x)) / g(x)^2.
So, differentiating h(x) = f(x)/g(x), we get h'(x) = (f'(x)*g(x) - f(x)*g'(x)) / g(x)^2
Substituting the given values, h'(2) = (f'(2)*g(2) - f(2)*g'(2)) / g(2)^2 = (-2*4 - (-3)*7) / 4^2 = (-8 + 21) / 16 = 13 / 16
D. h(x) = g(x)/ (1 + f(x))
To find h'(x), we need to use the quotient rule again.
Differentiating h(x) = g(x)/ (1 + f(x)), we get h'(x) = (g'(x)*(1 + f(x)) - g(x)*f'(x)) / (1 + f(x))^2
Substituting the given values, h'(2) = (g'(2)*(1 + f(2)) - g(2)*f'(2)) / (1 + f(2))^2 = (7*(1 + (-3)) - 4*(-2)) / (1 + (-3))^2 = (-7 + 8) / (-2)^2 = 1 / 4
Therefore, the correct answer is C. h(x) = f(x)/g(x), and h'(2) = 13/16.