I know that I have to apply the product or quotient rules for these problems but I'm not sure why I'm getting them wrong.
Please help me!
Find the derivative:
1) f(x) = x^6/108 times (6 ln(x) − 1)
I got: 6x^5 times 6 ln(x)-1 + x^6/108 times 6/x
2) f(x) = (ax + b)/(cx + k)
I got: x(cx+k) - (ax+b)x all over (cx+k) squared
3) y = 3x(ln x + ln 3) − 10x + e
I got: 3lnx + 3x/x - 10
1. f(x) = ((1/108)x^6)(6lnx - 1)
product rule:
f ' (x) = (1/108)x^6 (6/x) + (1/18)x^5 (6lnx - 1)
= (1/18)x^5 + (1/18)x^5)(6lnx - 1)
common factor of (1/18)x^5
= (1/18)x^5 (1 + 6lnx - 1)
= (1/3) x^5 lnx
check:
http://www.wolframalpha.com/input/?i=derivative+%28%281%2F108%29x%5E6%29%286lnx+-+1%29
2. y = (ax+b)/(cx+k) , where a, b,c, and k are constants
y' = ( a(cx+k) - c(ax+b) )/(cx+k)^2
= (acx + ak - acx - bc)/(cx+k)^2
= (ak - bc)/(cx+k)^2
3.
y = 3xlnx + 3ln3 x - 10x + e
y' = 3x(1/x) + 3lnx + 3ln3 - 10
= 3 + 3lnx + 3ln3 - 10
= 3lnx + 3ln3 - 7
= 3(lnx + ln3) - 7
= 3ln(3x) - 7
looks like you have some major studying ahead of you
Oh wow, I was really off. Thank you so much for explaining it to me!
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.