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Natasha_Volkova [10]
3 years ago
9

Can you give me a sentence for each if these words (15 points and brainliest to first answer)-

Mathematics
1 answer:
IrinaK [193]3 years ago
3 0
It is with trepidation that I accuse you of overusing Brainly's resources when you ask for help with 15 different items at one time.  I don't want to offend you (which is where trepidation comes into the picture), but need to tell this as I see it.

In the past, good dictionaries would give examples of word use.  Could you not ask your local librarian whether such a book is available?  or simply type the word into your google search box; Google will give examples of the use of that word.

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Y=x.arctan(x)^1/2. find dy/dx. pls show steps​
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y=x(\arctan x)^{1/2}

Use the product rule first:

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dx}{\mathrm dx}(\arctan x)^{1/2}+x\dfrac{\mathrm d(\arctan x)^{1/2}}{\mathrm dx}

\dfrac{\mathrm dy}{\mathrm dx}=(\arctan x)^{1/2}+x\dfrac{\mathrm d(\arctan x)^{1/2}}{\mathrm dx}

Use the chain rule to compute the derivative of (\arctan x)^{1/2}. Let z=(\arctan x)^{1/2} and take u=\arctan x, so that by the chain rule

\dfrac{\mathrm dz}{\mathrm dx}=\dfrac{\mathrm dz}{\mathrm du}\dfrac{\mathrm du}{\mathrm dx}

\dfrac{\mathrm dz}{\mathrm du}=\dfrac{\mathrm du^{1/2}}{\mathrm du}=\dfrac12u^{-1/2}

\dfrac{\mathrm du}{\mathrm dx}=\dfrac{\mathrm d\arctan x}{\mathrm dx}=\dfrac1{1+x^2}

\implies\dfrac{\mathrm d(\arctan x)^{1/2}}{\mathrm dx}=\dfrac1{2(\arctan x)^{1/2}(1+x^2)}

So we have

\dfrac{\mathrm dy}{\mathrm dx}=(\arctan x)^{1/2}+\dfrac x{2(\arctan x)^{1/2}(1+x^2)}

You can rewrite this a bit by factoring (\arctan x)^{-1/2}, just to make it look neater:

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac1{2(\arctan x)^{1/2}}\left(2\arctan x+\dfrac x{1+x^2}\right)

3 0
3 years ago
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