To determine why you are getting the wrong answers, let's review the correct application of the product and quotient rules.
1) f(x) = (x^6/108) * (6 ln(x) − 1)
To differentiate this function using the product rule, you need to differentiate each term separately and then apply the product rule formula. Let's break it down step by step:
First, differentiate the first term (x^6/108):
The power rule states that the derivative of x^n is nx^(n-1). Therefore, the derivative of x^6 is 6x^5.
Next, differentiate the second term (6 ln(x) − 1):
The derivative of ln(x) is 1/x using the chain rule. Therefore, the derivative of 6 ln(x) is 6/x.
The derivative of -1 is 0 since it is a constant.
Now, apply the product rule:
The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is given by [u'(x) * v(x)] + [u(x) * v'(x)].
In our case, u(x) = x^6/108 and v(x) = (6 ln(x) − 1).
So, u'(x) = 6x^5 and v'(x) = 6/x.
Now, apply the product rule formula:
f'(x) = [u'(x) * v(x)] + [u(x) * v'(x)]
= (6x^5 * (6 ln(x) − 1)) + ((x^6/108) * (6/x))
Simplifying this expression, you should get:
f'(x) = 36x^5 ln(x) - 6x^5 + x^5/18
So, your answer for the first problem is incorrect.
Let's move on to the second problem:
2) f(x) = (ax + b)/(cx + k)
To differentiate this function using the quotient rule, you need to differentiate the numerator and denominator separately and then apply the quotient rule formula.
Let's break it down step by step:
First, differentiate the numerator (ax + b):
The derivative of ax is a since b is a constant.
The derivative of b is 0 since it is a constant.
Next, differentiate the denominator (cx + k):
The derivative of cx is c since k is a constant.
The derivative of k is 0 since it is a constant.
Now, apply the quotient rule:
The quotient rule states that if you have two functions u(x) and v(x), then the derivative of their quotient is given by [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]^2.
In our case, u(x) = (ax + b) and v(x) = (cx + k).
So, u'(x) = a and v'(x) = c.
Now, apply the quotient rule formula:
f'(x) = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]^2
= [(a * (cx + k)) - ((ax + b) * c)] / (cx + k)^2
Simplifying this expression, you should get:
f'(x) = (ak - bc) / (cx + k)^2
So, your answer for the second problem is also incorrect.
Moving on to the third problem:
3) y = 3x(ln x + ln 3) − 10x + e
To find the derivative of this function, you need to differentiate each term separately.
Let's break it down step by step:
The derivative of 3x is simply 3.
The derivative of (ln x + ln 3) can be found using the chain rule. The derivative of ln x is 1/x, and the derivative of ln 3 is 0 since it is a constant. Therefore, the derivative of (ln x + ln 3) is 1/x.
The derivative of -10x is -10.
The derivative of the constant term e is 0 since it is a constant.
Putting it all together, the derivative of y with respect to x is:
y' = 3 + 1/x - 10 + 0
= -7 + 1/x
So, your answer for the third problem is also incorrect.
Please double-check your calculations and make sure to properly apply the product and quotient rules as explained above.