dy\dx=x+1\square rod of x
where y=5
x= 4
from chapter 5 (integration)
This question makes no sense to me.
Perhaps you want to know y(x), knowing the derivative
dy/dx = x + (1/sqrt x)
However, I am not sure from the way you typed it whether you mean this instead:
dy/dx = (x+1)/sqrt x
In either case, integrate to get y(x) with an unknown constant term.
Use the fact that y = 5 when x = 4 to get the constant.
In the first case, the integral is
y = x^2/2 -(1/2)*x^-3/2 + C
To find the value of dy/dx for the given equation, we need to differentiate the equation with respect to x.
The given equation is:
dy/dx = (x + 1)^(square root of x)
In order to differentiate this equation, we will use the chain rule. The chain rule states that if we have a composite function, f(g(x)), the derivative of the composite function is given by:
d/dx [ f(g(x)) ] = f'(g(x)) * g'(x)
Applying the chain rule to the given equation, we have:
dy/dx = d/dx [ (x + 1)^(square root of x) ]
To differentiate (x + 1)^(square root of x), we can rewrite it as e^(ln(x+1) * sqrt(x)), where e is the base of natural logarithm. Now we can use the chain rule to differentiate e^(ln(x+1) * sqrt(x)).
Let's break it down:
1. Find the derivative of ln(x+1) * sqrt(x) with respect to x.
For this, we can use the product rule, which states that if we have two functions, f(x) and g(x), the derivative of their product is given by:
(f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x)
Apply the product rule to ln(x+1) * sqrt(x):
d/dx [ ln(x+1) * sqrt(x) ] = (1 / (x+1)) * sqrt(x) + ln(x+1) * (1/(2 * sqrt(x)))
2. Now, let's find the derivative of e^(ln(x+1) * sqrt(x)) with respect to x.
We can use the chain rule here. Let u = ln(x+1) * sqrt(x), and apply the chain rule:
d/dx [ e^(ln(x+1) * sqrt(x)) ] = e^u * u'
Substituting u = ln(x+1) * sqrt(x) and u' = (1 / (x+1)) * sqrt(x) + ln(x+1) * (1/(2 * sqrt(x))), we get:
dy/dx = e^(ln(x+1) * sqrt(x)) * [(1 / (x+1)) * sqrt(x) + ln(x+1) * (1/(2 * sqrt(x)))]
Simplifying further, we have:
dy/dx = (x + 1)^(square root of x) * [(1 / (x+1)) * sqrt(x) + ln(x+1) * (1/(2 * sqrt(x)))]
Now, substitute the given values y = 5 and x = 4 into the equation to find the value of dy/dx:
dy/dx = (4 + 1)^(square root of 4) * [(1 / (4+1)) * sqrt(4) + ln(4+1) * (1/(2 * sqrt(4)))]
dy/dx = 5^(2) * [(1/5) * 2 + ln(5) * (1/4)]
dy/dx = 25 * [0.4 + ln(5) * 0.25]
dy/dx ≈ 25 * [0.4 + 0.25 * ln(5)] ≈ 25 * (0.4 + 0.25 * 1.609) ≈ 25 * (0.4 + 0.40225) ≈ 25 * 0.80225 ≈ 20.05625
Therefore, dy/dx (given y=5 and x=4) is approximately 20.05625.