If h(x) = Squareroot 7 + 6f(x), where f(1) = 7 and f'(1) = 2, find h'(1)
h(x) = √7 + 6f(x)
Now that seems a bit ridiculous, so I will go with
h(x) = √(7+6f(x))
by the chain rule,
h'(x) = 1/(2√(7+6f(x)) * 6f'(x)
so h'(1) = 1/(2√(7+6*7)) * 6*2 = 1/(2*7) * 12 = 6/7
ahh thank you!! The square root was kind of throwing me off
To find the derivative of h(x), which is denoted as h'(x), we can use the chain rule.
The chain rule states that if we have a composite function, f(g(x)), then the derivative of f(g(x)) with respect to x is equal to the derivative of f with respect to g, multiplied by the derivative of g with respect to x.
In this case, h(x) is a composite function of f(x) and g(x) = 7 + 6f(x). So, we need to find the derivative of h(x) by applying the chain rule.
First, let's find the derivative of g(x) = 7 + 6f(x). Since f(x) is given, we can differentiate g(x) with respect to x.
g'(x) = d(7 + 6f(x))/dx.
The derivative of a constant term (7, in this case) with respect to x is zero, so it disappears.
g'(x) = d(6f(x))/dx.
Now we need to use the chain rule to differentiate the composite function 6f(x). Since f(x) is a function of x, we can rewrite it as 6f(u), where u = x.
Using the chain rule, the derivative of 6f(u) with respect to u is 6 * f'(u).
Therefore, g'(x) = 6 * f'(x).
Now we know the derivative of g(x), which is g'(x) = 6 * f'(x).
To find the derivative of h(x) = sqrt(7 + 6f(x)), we apply the chain rule again:
h'(x) = d(sqrt(7 + 6f(x)))/dx.
Using the chain rule, the derivative of sqrt(7 + 6f(x)) with respect to x is equal to the derivative of sqrt(7 +6f(x)) with respect to 7 + 6f(x), multiplied by the derivative of 7 + 6f(x) with respect to x.
The derivative of sqrt(u) with respect to u can be found using the power rule for derivatives:
d(sqrt(u))/du = (1/2) * u^(-1/2).
Applying this to our equation, the derivative of sqrt(7 + 6f(x)) with respect to 7 + 6f(x) is:
(1/2) * (7 + 6f(x))^(-1/2).
The derivative of 7 + 6f(x) with respect to x is g'(x), which we found earlier to be 6 * f'(x).
Putting it all together, we have:
h'(x) = (1/2) * (7 + 6f(x))^(-1/2) * 6 * f'(x).
Now, we are given that f(1) = 7 and f'(1) = 2. So, we can evaluate h'(1) by substituting these values into the derived equation:
h'(1) = (1/2) * (7 + 6f(1))^(-1/2) * 6 * f'(1).
h'(1) = (1/2) * (7 + 6*7)^(-1/2) * 6 * 2.
h'(1) = (1/2) * (7 + 42)^(-1/2) * 12.
h'(1) = (1/2) * (49)^(-1/2) * 12.
h'(1) = (1/2) * (1/7) * 12.
h'(1) = 6/7.
Therefore, h'(1) = 6/7.