Find the derivative of f(x) = 1/x(27,000,000/pi + 6,750,000) + 20pix^2
Can someone help me with this? Thanks.
I am not certain if the 20pi x^2 is in the denominator or not of the first term. If it is really this,
f(x)=[1/(x(27,000,000/pi + 6,750,000)] + 20PI x^2
then work it in two parts, the first, and second term.
f(x)=1/kx + 20 pi x^2
f'= -1/kx^2 + 40PI x
where k= 27E6/PI +6.75E6
Here I'll write it again but distribute the 1/x so you can see it better.
f(x) = (27,000,000/pix)+(6,750,000/x)+20pix^2
Does that help?
The 1/x was only factored out of the first two terms, not out of the
20pi(x^2)
But I get the general idea. I just write 1/x as x^-1 and differentiate that. Makes sense, thanks!
Sure! To find the derivative of the function f(x) = 1/x(27,000,000/pi + 6,750,000) + 20πx^2, we can use the basic rules of differentiation.
First, let's break down the function into its two terms:
f(x) = 1/x(27,000,000/pi + 6,750,000) + 20πx^2
The derivative of the first term, 1/x(27,000,000/pi + 6,750,000), can be found using the quotient rule.
The quotient rule states that if we have a function in the form f(x) = g(x)/h(x), then the derivative is given by:
f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
Let's denote g(x) = 1 and h(x) = x(27,000,000/pi + 6,750,000).
Now, let's find the derivatives of g(x) and h(x):
g'(x) = 0 (since g(x) = 1, the derivative of a constant is always 0)
h'(x) = 1(27,000,000/pi + 6,750,000) - x(0) = 27,000,000/pi + 6,750,000
Now, let's substitute the values into the quotient rule formula:
f'(x) = (0 * x(27,000,000/pi + 6,750,000) - 1 * (27,000,000/pi + 6,750,000)) / (x(27,000,000/pi + 6,750,000))^2
Simplifying further:
f'(x) = -(27,000,000/pi + 6,750,000) / (x(27,000,000/pi + 6,750,000))^2
Moving on to the second term, 20πx^2, we can use the power rule to find its derivative.
The power rule states that if we have a function in the form f(x) = ax^n, then the derivative is given by:
f'(x) = n * ax^(n-1)
In this case, a = 20π and n = 2. Let's apply the power rule:
f'(x) = 2 * 20πx^(2-1)
Simplifying further:
f'(x) = 40πx
Now we have found the derivatives of both terms in the original function.
To find the derivative of the entire function f(x), we can simply add the derivatives of the individual terms together:
f'(x) = -(27,000,000/pi + 6,750,000) / (x(27,000,000/pi + 6,750,000))^2 + 40πx
And that's the derivative of f(x) = 1/x(27,000,000/pi + 6,750,000) + 20πx^2.