differential calculas :
x^2 − 49 / 7 − x
To find the derivative of the given function, we can use the quotient rule. The quotient rule states that if we have a function f(x)=g(x)/h(x), then its derivative f'(x) can be calculated as (g'(x)h(x) - g(x)h'(x))/[h(x)]^2.
Let's find the derivative of the given function step by step:
Function: f(x) = (x^2 - 49)/(7 - x)
First, let's find the derivative of the numerator: g(x) = x^2 - 49.
g'(x) = 2x.
Next, let's find the derivative of the denominator: h(x) = 7 - x.
h'(x) = -1.
Now, let's substitute these values into the quotient rule formula:
f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2
= [(2x)(7 - x) - (x^2 - 49)(-1)]/(7 - x)^2
= (14x - 2x^2 + x^2 - 49)/(7 - x)^2
= (-x^2 + 14x - 49)/(7 - x)^2.
So, the derivative of the given function is f'(x) = (-x^2 + 14x - 49)/(7 - x)^2.