Hello,
Could somebody please help me with the following question? It asks to differentiate the function below according to derivate rules of calculus such as the power rule (if f(x)=x^n, then f'(x)=nx^n-1), the product rule (if F(x)=f(x)g(x) then F'(x)=f(x)g'(x)+f'(x)g(x)) and the chain rule for polynomials (if F(x)=(f(x))^n then F'(x)=nf'(x)f(x)^n-1).
Here's the function:
f(x)=3x/x^2+4
My tentative solution:
f(x)=3x/x^2+4
f(x)=(3x)(x^2+4)^-2
f'(x)=(3x)(-2)(x^2+4)(x^2+4)^-2-1 (chain rule)
f'(x)=(3x)(-2)(x^2+4)(x^2+4)^-3
f'(x)=(-6x)(x^2+4)(x^2+4)^-3
I'm not sure whether to apply the chain rule of the product rule in step 2 or both of them.
Any help would be much appreciated!
Constantin
To find the output of f(-3), we need to plug in -3 for x in the function rule f(x):
f(-3) = 2(-3)^2 - 7(-3) + 1
Simplifying the expression:
f(-3) = 18 + 21 + 1
f(-3) = 40
Therefore, the output of f(-3) is 40.
You're welcome! I'm glad I could help.
you need both the product rule and the chain rule.
f(x) = (3x)(x^2+4)^-1
f'(x) = (3)(x^2+4)^-1 + (3x)(-1)(x^2+4)^-2(2x)
= 3/(x^2+4) - 6x^2/(x^2+4)^2
= (3(x^2+4)-6x^2)/(x^2+4)^2
= (12-3x^2)/(x^2+4)^2
The quotient rule produces the same result; in fact, you can see the intermediate steps in the calculations above, if you look carefully.
Thank you Steve! I understand it now.
Given the function rule f(x) = 2x ^ 2 - 7x + 1 what is the
output of f(- 3) :
Thanks u a life saver
Hi Constantin! I'll be happy to help you with the differentiation of the function f(x) = 3x/(x^2 + 4).
To differentiate this function, you need to apply both the quotient rule and the chain rule. Let me guide you through the steps:
Step 1: Identify the numerator and denominator.
The numerator is 3x, and the denominator is (x^2 + 4).
Step 2: Apply the quotient rule.
The quotient rule states that if you have a function f(x) = g(x)/h(x), then its derivative f'(x) is given by (g'(x)h(x) - g(x)h'(x))/[h(x)]^2.
Using the quotient rule, we have:
f'(x) = [(3)(x^2 + 4) - (3x)(2x)] / [(x^2 + 4)^2]
Step 3: Simplify the expression.
Expanding and simplifying the numerator gives us:
f'(x) = (3x^2 + 12 - 6x^2) / (x^2 + 4)^2
= (12 - 3x^2) / (x^2 + 4)^2
So, the derivative of f(x) = 3x/(x^2 + 4) is f'(x) = (12 - 3x^2) / (x^2 + 4)^2.
You did a great job recognizing that the chain rule was necessary. However, in the second step, you actually need to apply the quotient rule instead of the chain rule. The chain rule is applicable when you have a composite function, involving the composition of multiple functions.
Remember to check your work by simplifying the expression and ensuring that it cannot be further simplified.
I hope this clarifies the steps to differentiate the function! Let me know if you have any further questions.