hey, i'd really appreciate some help differentiating this equation:
t= (4+x^2)^(1/2) /3 + (3-x)/5
so that's four plus x to the power of 2 all to the power of 1/2 all over three plus 3-x divided by five... :P
Thankyou.
I would rewrite it as
t= (1/3)(4+x^2)^(1/2) + 3/5 - (1/5)x
then
dt/dx = (1/6)(4+x^2)^(-1/2)(2x) - 1/5
=(1/3)x((4+x^2)^(-1/2) - 1/5
At this point, further simplification could be done, depending on the demands of the teacher or the textbook.
To differentiate the given equation, we will use the rules of differentiation. Here's how you can differentiate it step by step:
Step 1: Rewrite the equation as t = (1/3)(4 + x^2)^(1/2) + (3 - x)/5.
Step 2: Differentiate each term in the equation separately.
For the first term, (1/3)(4 + x^2)^(1/2), we need to use the chain rule.
The chain rule states that if we have a function inside a function, like f(g(x)), the derivative of f(g(x)) is given by f'(g(x)) * g'(x).
In this case, the outer function is (1/3)(4 + x^2)^(1/2), and the inner function is (4 + x^2)^(1/2). We differentiate the outer function as (1/3), and the inner function as (4 + x^2)^(-1/2) * 2x using the power rule and the chain rule.
So, the derivative of the first term is (1/3) * (4 + x^2)^(-1/2) * 2x.
For the second term, (3 - x)/5, we can simply differentiate it as (-1/5) since the derivative of a constant multiplied by x is equal to the constant itself.
Step 3: Combine the derivatives of each term to obtain the final derivative.
The derivative of t with respect to x (dt/dx) is the sum of the derivatives of each term:
dt/dx = (1/3) * (4 + x^2)^(-1/2) * 2x - 1/5.
The equation can be further simplified depending on the demands of the teacher or the textbook.