To determine
To find: f'x by definition of derivative.
Answer
f'x= -52 3-5x
Explanation
1) Concept:
By using definition of derivative in terms of limit we can find derivative of given function.
2) Formula:
Derivative of function f at x is denoted by f'x, and is given by,
f'x=limh → 0fx+h-fxh
if limit exists.
3) Given:
fx=3-5x
4) Calculations:
As fx=3-5x
f'x=limh→ 0fx+h-fxh
= limh→ 03-5x+h-3-5xh
=limh→ 03-5x+h-3-5xh . 3-5x+h+3-5x3-5x+h+3-5x
=limh→ 03-5x+h-3-5xh 3-5x+h+3-5x
=limh→ 03-5x+h-(3-5x)h (3-5x+h+3-5x)
=limh→ 03-5x-5h -3+5xh (3-5x+h+3-5x)
=limh→ 0-5h h (3-5x+h+3-5x)
=limh→ 0-53-5x+h+3-5x
= -5(3-5x+0+3-5x)
f'x= -52 3-5x
Conclusion:
f'x= -52 3-5x
To determine
To find: Domain of f and f’.
Answer
The domain of given function f is {x| x ≤ 3/5 }.
In interval notation -∞,35.
The domain of function f ’ is {x| x < 3/5 }.
In interval notation -∞,35.
Explanation
1) Concept:
The domain of the function is set of all such t for which function is defined.
That is for finding domain of given function we have to exclude points where function is undefined.
Because square root of negative number is not defined.
Hence Domain of ft= gt, is domain of g(t) ≥ 0 .
As given function is rational function, it is undefined where denominator becomes zero.
As there is nth root where n is even in denominator of the form gx, then function h(x) is defined for all x for which g(x) > 0 .
.
2) Given:
fx=3-5x
f'x= -52 3-5x
3) Calculations:
Since x=3-5x, f(x) exist when 3-5x≥0, adding 5x on both sides we get
3≥5x so x≤35
Therefore, domain of given function f is {x| x ≤ 3/5 }.
In interval notation -∞,35.
Since f'x= -52 3-5x,
Since this is rational function including square root in denominator, we need quantity inside the square root is positive as well as non-zero.
So, f’(x) exist when 3-5x>0, adding 5x on both sides we get
3>5x so x<35
Therefore, domain of given function f is {x| x <3/5 }.
In interval notation -∞,35.
Conclusion:
Domain of f -∞,35.
Domain of f’-∞,35.
To determine
To sketch: Graph of f and f'.
Answer
Explanation
From graph it is seen that f'x is always negative as f(x) is decreasing function.
The answer from part ‘a’ is reasonable since it is negative for any value of x.
