Find the first and second derivative of the function:
x/(7x+10)
Using quotient rule. This is part of my homework in the section named "Higher Order Derivatives". I am confused on getting the second derivative, as I found the first derivative to be 10/(7x+10)^2
I agree with your first derivative.
Now, start with 10/(7x+10)^2 and take the derivative again.
Denom times deriv of numerator = 0
since deriv of 10 = 0
minus the numer time the deriv of the denomin.
- 10(2(7x+10)7) this is the chain rule
You need to simplify this expression
all over the denominator squared which makes the denominator to the 4th power.
y=x/(7x+10)
y'=1/( ) - 7x/( )^2
= (7x+10)x - 7x)/ (7x+10)^2
and that is not equal to your answer.
= (7x^2 +3x)/(7x+10)^2
To find the first derivative of a function using the quotient rule, you need to differentiate the numerator and denominator separately.
Let's begin by applying the quotient rule to find the first derivative of the function f(x) = x/(7x+10).
First, differentiate the numerator:
f'(x) = (1)(7x+10) - x(7)/(7x+10)^2
Next, simplify and combine like terms:
f'(x) = 7x + 10 - 7x/(7x+10)^2
Simplifying further, we get:
f'(x) = 10/(7x+10)^2
Now, to find the second derivative, you need to differentiate the first derivative you just found.
Differentiate f'(x) = 10/(7x+10)^2 using the quotient rule.
First, differentiate the numerator:
f''(x) = (0)(7x+10)^2 - 10(2)(7x+10)(7)/(7x+10)^4
Next, simplify and combine like terms:
f''(x) = -140(7x+10)/(7x+10)^4
Simplifying further, we get:
f''(x) = -140/(7x+10)^3
Therefore, the first derivative is 10/(7x+10)^2, and the second derivative is -140/(7x+10)^3.
Remember, when using the quotient rule, the first derivative is obtained by differentiating the numerator and denominator separately, while the second derivative is found by differentiating the first derivative.