how to differentiate 500x^2 [(280+x^2)^-1/2] +10x?
let y = (500x^2)(280 + x^2)^(-1/2) + 10x
dy/dx = 500x^2(-1/2)(280 + x^2)^(-3/2) (2x) + 1000x(280 + x^2)^(-1/2) + 10
this is your "first-line" derivative.
I don't know to what level of simplification you have to take it.
i got to that step and got stuck, but cant seem to simplify it correctly. Thanks for your feedback! Appreciate it :)
oh nvm, I got it. i didn't isolate the 10. Thanks!
To differentiate the given expression, which is 500x^2 [(280+x^2)^-1/2] + 10x, we will use the rules of differentiation step by step. Let's break it down:
Step 1: Apply the product rule
The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product, u(x) * v(x), is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
In our expression, u(x) is 500x^2 and v(x) is [(280+x^2)^-1/2].
So, applying the product rule, we have:
(d/dx)(500x^2 [(280+x^2)^-1/2]) = (d/dx)(500x^2) * [(280+x^2)^-1/2] + 500x^2 * (d/dx)([(280+x^2)^-1/2])
Step 2: Differentiate u(x) = 500x^2
The derivative of u(x) = 500x^2 is obtained using the power rule of differentiation. According to the power rule, if we have a function of the form f(x) = ax^n, then its derivative is given by:
(d/dx)(ax^n) = nax^(n-1)
In our case, n = 2 and a = 500. Therefore, differentiating 500x^2, we get:
(d/dx)(500x^2) = 2 * 500 * x^(2-1) = 1000x
Step 3: Differentiate v(x) = [(280+x^2)^-1/2]
To differentiate v(x), we need to apply the chain rule. The chain rule states that if we have a composition of functions, y = f(g(x)), then the derivative of y with respect to x is given by:
(d/dx)(f(g(x))) = f'(g(x)) * g'(x)
In our case, f(u) = u^(-1/2) and g(x) = 280 + x^2. Let's differentiate v(x):
(d/dx)([(280+x^2)^-1/2]) = (-1/2) * (280 + x^2)^(-1/2 - 1) * (d/dx)(280 + x^2)
To find (d/dx)(280 + x^2), we apply the power rule:
(d/dx)(280 + x^2) = 2x
So,
(d/dx)([(280+x^2)^-1/2]) = (-1/2) * (280 + x^2)^(-1/2 - 1) * 2x
= -x / (280 + x^2)^(3/2)
Step 4: Substitute the derivatives back into the product rule equation
Now that we have the derivatives of u(x) and v(x), we can substitute them back into the product rule equation we derived in Step 1:
(d/dx)(500x^2 [(280+x^2)^-1/2]) = 1000x * [(280+x^2)^-1/2] + 500x^2 * (-x / (280 + x^2)^(3/2))
Step 5: Simplify the expression
To simplify the expression, we can factor out common terms and combine like terms if possible. However, without knowing the specific values of x, it is difficult to simplify further.
So, after differentiating the given expression, we have:
(d/dx)(500x^2 [(280+x^2)^-1/2] + 10x) = 1000x * [(280+x^2)^-1/2] + 500x^2 * (-x / (280 + x^2)^(3/2)) + 10
This is the differentiation of the given expression.