Find using the defintion f '(c) lim = f(x) - f(c)
x -> c _______
x - c
the gradient function f ' (x) for:
a) f(x) = 1/x
b)f(x) = √ x
[My tutor told me to rationalize before doing the question for part b but how do i rationalize this]
Find using the defintion f '(c) lim = f(x) - f(c)
x -> c _______
x - c
the gradient function f ' (x) for:
a) f(x) = 1/x
b)f(x) = √ x
[My tutor told me to rationalize before doing the question for part b but how do i rationalize this]
Hi
To rationalize the numerator of $\displaystyle \frac{\sqrt{x}-\sqrt{c}}{x-c}$, multiply and divide the fraction by the conjugate of $\displaystyle \sqrt{x}-\sqrt{c}$ which is $\displaystyle \sqrt{x}+\sqrt{c}$ :
$\displaystyle \frac{\sqrt{x}-\sqrt{c}}{x-c}=\frac{(\sqrt{x}-\sqrt{c})(\sqrt{x}+\sqrt{c})}{(x-c)(\sqrt{x}+\sqrt{c})}=\ldots$
(notice that the numerator looks like $\displaystyle (a-b)(a+b)$ which equals $\displaystyle a^2-b^2$)
Edit : for the first question : $\displaystyle \frac{f(x)-f(c)}{x-c}=\frac{\frac{1}{x}-\frac{1}{c}}{x-c}=\frac{\frac{c-x}{xc}}{x-c}=\frac{c-x}{xc(x-c)}=-\frac{1}{xc}$ hence $\displaystyle f'(c) = ?$
Sorry I didn't realize that the definition for f '(c) was all over the place....sorry about that.
For Part b is the answer:
(x-c)/(x√x + x√c - c√x - c√c
and for Part (a) how am I supposed to find f '(x)
Sorry this question is really confusing me
coz there are so many x's and c's.......
That's it but if you try computing $\displaystyle f'(c)=\lim_{x\to c}\frac{x-c}{x\sqrt{x}+x\sqrt{c}-c\sqrt{x}-c\sqrt{c}}$ you'll see that this limit is an indeterminate form. (both the numerator and the denominator tend to 0) If you hadn't expanded the numerator, it would have given you
$\displaystyle \frac{\sqrt{x}-\sqrt{c}}{x-c}=\frac{(\sqrt{x}-\sqrt{c})(\sqrt{x}+\sqrt{c})}{(x-c)(\sqrt{x}+\sqrt{c})}=\frac{x-c}{(x-c)\left(\sqrt{x}+\sqrt{c}\right)}=\frac{1}{\sqrt{x }+\sqrt{c}}$
And $\displaystyle f'(c)=\lim_{x\to c}\frac{1}{\sqrt{x}+\sqrt{c}}$ isn't an indeterminate form any longer. Can you find the value of this limit ?
The gradient of $\displaystyle f$ taken at $\displaystyle c$ is $\displaystyle f'(c)=\lim_{x\to c} -\frac{1}{xc}= -\frac{1}{c\cdot c}=-\frac{1}{c^2}$ as long as $\displaystyle c\neq 0$. Thus $\displaystyle \boxed{f'(x)=-\frac{1}{x^2}}$ for $\displaystyle x\neq 0$.and for Part (a) how am I supposed to find f '(x)
Sorry this question is really confusing me
coz there are so many x's and c's.......