Let f(x)=–5x(x–5).
Then f'(–1)=
And after simplifying f'(x)=
Hint: You may want to expand and simplify the expression for f(x) first.
John, I really believe that anybody who studies Calculus should be able to do an easy question such as this.
f(x) = 5x^2 - 25x
I will let you take it from there
To find the derivative of the function f(x) = -5x(x-5), we can use the product rule. The product rule states that if we have a function h(x) = f(x) * g(x), then the derivative of h(x) is given by h'(x) = f'(x) * g(x) + f(x) * g'(x).
First, let's expand and simplify the expression for f(x):
f(x) = -5x(x-5)
= -5x^2 + 25x
Now, to find f'(x), we can apply the product rule. Let f(x) = -5x^2 and g(x) = x-5.
f'(x) = f'(x) * g(x) + f(x) * g'(x)
To find f'(x), we differentiate f(x) with respect to x.
f'(x) = -10x
Next, to find g'(x), we differentiate g(x) = x-5 with respect to x.
g'(x) = 1
Now we can substitute the values into the product rule formula:
f'(x) = f'(x) * g(x) + f(x) * g'(x)
= (-10x) * (x-5) + (-5x^2) * 1
= -10x^2 + 50x - 5x^2
= -15x^2 + 50x
Now, we can find f'(-1) by substituting x = -1 into the expression for f'(x):
f'(-1) = -15(-1)^2 + 50(-1)
= -15 + 50
= 35
Therefore, f'(-1) = 35.
After simplifying f'(x), we get f'(x) = -15x^2 + 50x.