Using the definition of derivative find f'(x) when f(x) = (1/sqr x) + x.
f(x+h) = 1/√(x+h) + x+h
f(x) = 1/√x + x
f(x+h)-f(x) = 1/√(x+h) - 1/√x + h
= (x+h)^(-1/2) - x^(-1/2) + h
From the binomial theorem,
(a+b)^n = a^n + n a^n-1 b + ...
(x+h)^(-1/2) = x^(-1/2) - 1/2 x^(-3/2)h + 3/8 x^(-5/2)h^2 + ...
where ... is stuff with higher powers of h.
Now, we just subtract
f(x+h)-f(x) = h -1/2 x^(-3/2)h + 3/8 x^(-5/2)h^2 + ...
Now for the quotient
(f(x+h)-f(x))/h = 1 - 1/2 x^-3/2 + 3/8 x^(-5/2)h + ...
Now take the limit as h->0 and it all vanishes except
f'(x) = 1 - 1/2 x^-3/2
sometimes called:
find the derivative from First Priciples
f(x) = 1/√x + x
f(x+h) = 1/√(x+h) + x+h
f'(x) = Lim ( f(x+h) - f(x) )/h , as h --->0
= lim( 1/√(x+h) + x+h - (1/√x + x )/h as h--->0
= lim ( (1/√(x+h) - 1/√x)/h + h/h) as h--> 0
= lim ( (1/√(x+h) - 1/√x)/h* (√x√(x+h)/(√x√x+h) + h/h) as h--> 0
= Lim ( (√x - √(x+h)/(h√x√(x+h) ) + h/h) as h --->0
= Lim ( (√x - √(x+h)/(h√x√(x+h)*(√x + √(x+h)/(√x + √(x+h)) ) + h/h) as h --->0
= lim (x - x - h)/(h√x√(x+h)(√x + √(x+h)) + h/h
= lim -1/(√x√(x+h)(√x + √(x+h) ) + 1 , as h --->0
= -1/(√x√x(√x + √x) + 1
= -1/(x(2√x)) + 1
or -1/(2(√x)^3)
Wheewww!
Darn!!! last line, forgot the +1 , but you figured that out, didn't you?
To find the derivative of the function f(x) = (1/sqrt(x)) + x using the definition of the derivative, we need to compute the following:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Step 1: Evaluate f(x + h)
To find f(x + h), we substitute (x + h) into the function f(x):
f(x + h) = (1/sqrt(x + h)) + (x + h)
Step 2: Evaluate f(x)
Now let's find f(x) by substituting x into the function f(x):
f(x) = (1/sqrt(x)) + x
Step 3: Substitute the results into the derivative formula
Substituting the results from Step 1 and Step 2 into the derivative formula:
f'(x) = lim(h->0) [(1/sqrt(x + h)) + (x + h) - ((1/sqrt(x)) + x)] / h
Simplify the expression:
f'(x) = lim(h->0) [(1/sqrt(x + h)) - (1/sqrt(x))] / h
Step 4: Simplify the expression further
To simplify, let's get rid of the fractions in the numerator:
f'(x) = lim(h->0) [(sqrt(x) - sqrt(x + h)) / (sqrt(x) * sqrt(x + h))] / h
Now, combine the square roots in the numerator:
f'(x) = lim(h->0) [sqrt(x) - sqrt(x + h)] / (h * sqrt(x) * sqrt(x + h))
Step 5: Rationalize the denominator
To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator, which is
(sqrt(x) * sqrt(x + h)) + sqrt(x) * sqrt(x + h)
f'(x) = lim(h->0) [sqrt(x) - sqrt(x + h)] * [sqrt(x) * sqrt(x + h) + sqrt(x) * sqrt(x + h)] / (h * sqrt(x) * sqrt(x + h) * [sqrt(x) * sqrt(x + h) + sqrt(x) * sqrt(x + h)])
Now, simplify the expression further by multiplying out the numerator:
f'(x) = lim(h->0) [(x - (x + h)) / (h * sqrt(x) * sqrt(x + h) * [sqrt(x) * sqrt(x + h) + sqrt(x) * sqrt(x + h)])]
Simplify the numerator:
f'(x) = lim(h->0) (-h) / (h * sqrt(x) * sqrt(x + h) * [sqrt(x) * sqrt(x + h) + sqrt(x) * sqrt(x + h)])
Step 6: Cancel out the common factors
Cancel out the common factors in the numerator and denominator:
f'(x) = lim(h->0) -1 / (sqrt(x) * sqrt(x + h) * [sqrt(x) * sqrt(x + h) + sqrt(x) * sqrt(x + h)])
Simplify the expression further:
f'(x) = lim(h->0) -1 / (2 * sqrt(x) * [sqrt(x) * sqrt(x + h) + sqrt(x) * sqrt(x + h)])
Step 7: Evaluate the limit
As h approaches 0, the expression becomes:
f'(x) = -1 / (2 * sqrt(x) * 2 * sqrt(x))
Simplify further:
f'(x) = -1 / (4 * x)
Therefore, the derivative of the function f(x) = (1/sqrt(x)) + x is f'(x) = -1 / (4 * x).