Need help with differentiating a problem

Question

I am attempting to differentiate: $$f(x)= \dfrac{x^{0.5}}{2x^{1.5}}$$

i first simplified the fraction by subracting the exponemts leaving me with: $\frac{x}{2x}$

Then I simplified the fraction further giving me: $\frac{1}{x}$

using law of exponents i rearanged the fraction to: $x^{-1}$

Finally using the power rule, i differentiated to obtain: $-x^{-2}$

I got this wrong and i dont know where i went wrong with it, can someone help me point out my errors?

Thank you

Answer 1

it simplifies to $$\frac{1}{2x}$$ and the first derivative is given by $$\frac{1}{2}(-1)x^{-2}$$

Answer 2

The derivative of $x^{-1}$ is $-x^{-2}$ as you correctly pointed out. Now the function can be simplified to $$f(x) = \frac{1}{2}x^{-1}$$ So you just need to multiply the derivative of $x^{-1}$ by $1/2$: $$f'(x) =-\frac{1}{2}x^{-2}$$

ALSO, just to clarify how you should go about simplifying the exponents step by step: $$f(x)=\dfrac{x^{0.5}}{2x^{1.5}}$$ $$=\dfrac{1}{2x^{1.5}x^{-0.5}}$$ $$=\dfrac{1}{2x^{1.5-0.5}}$$ $$=\dfrac{1}{2x^{1}}$$ $$=\dfrac{1}{2x}$$

Answer 3

We have $$f(x)= \dfrac{x^{0.5}}{2x^{1.5}}=\frac12x^{0.5-1.5}=\frac12x^{-1}$$ so $$f'(x)=\frac12(-1)x^{-2}=-\frac1{2x^2}.$$