Asked by Ben
f(a) = a + √a
(a) Find the derivative of the function using the definition of derivative.
(a) Find the derivative of the function using the definition of derivative.
Answers
Answered by
drwls
The derivative f'(a) is the sum of the derivative of a and the derivative of sqrt a. In this case, a is treated as a variable, not a constant.
The answer is 1 + 1/(sqrt a)
The answer is 1 + 1/(sqrt a)
Answered by
Damon
f(a + h) = a + h + (a+h)^1/2
f(a) = a + a^(1/2)
f(a+h) - f(a) = h + (a+h)^1/2 - a^1/2
binomial series for q<p: (p+q)^n= p^n +n p^(n-1) q + n(n-1)/2! p^(n-2)q^2 ....
so
for small h
f(a+h)-f(a) = h + a^1/2 + (1/2) a^(-1/2)h + (1/2)(-1/2)/2*a^(-1.5)h^2.. -a^1/2)
f(a+h)-f(a) = h + (1/2)a^(-1/2) h - 1/8 a^-1.5 h^2 ....
divide by h
(f(a+h)-f(a))/h = 1 + (1/2) a^-1/2 - (1/8) a^-1.5) h ...
let h-->0
df/da---> 1 + (1/2)a^(-1/2)
f(a) = a + a^(1/2)
f(a+h) - f(a) = h + (a+h)^1/2 - a^1/2
binomial series for q<p: (p+q)^n= p^n +n p^(n-1) q + n(n-1)/2! p^(n-2)q^2 ....
so
for small h
f(a+h)-f(a) = h + a^1/2 + (1/2) a^(-1/2)h + (1/2)(-1/2)/2*a^(-1.5)h^2.. -a^1/2)
f(a+h)-f(a) = h + (1/2)a^(-1/2) h - 1/8 a^-1.5 h^2 ....
divide by h
(f(a+h)-f(a))/h = 1 + (1/2) a^-1/2 - (1/8) a^-1.5) h ...
let h-->0
df/da---> 1 + (1/2)a^(-1/2)
Answered by
drwls
whoops, I forgot the (1/2). Damon also provided the derivation you wanted.
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.