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DifferentiationQuestion and Answers: Page 1

Question Number 211508 Answers: 1 Comments: 3

If (√(1 − x^2 )) + (√(1 − y^2 )) = a(x − y) then prove that (dy/dx) = (√(((1 − y^2 )/(1 − x^2 )) )) .

$$\mathrm{If}\:\sqrt{\mathrm{1}\:−\:{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}\:−\:{y}^{\mathrm{2}} }\:=\:{a}\left({x}\:−\:{y}\right)\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\frac{{dy}}{{dx}}\:=\:\sqrt{\frac{\mathrm{1}\:−\:{y}^{\mathrm{2}} }{\mathrm{1}\:−\:{x}^{\mathrm{2}} }\:}\:. \\ $$

Question Number 211502 Answers: 2 Comments: 0

If { ((f(x)=x^2 )),((g(x)=sin x)) :}, Then find (df/dg).

$$\mathrm{If}\:\begin{cases}{{f}\left({x}\right)={x}^{\mathrm{2}} }\\{{g}\left({x}\right)=\mathrm{sin}\:{x}}\end{cases}, \\ $$$$\mathrm{Then}\:\mathrm{find}\:\frac{{df}}{{dg}}. \\ $$

Question Number 211365 Answers: 2 Comments: 0

Question Number 211321 Answers: 1 Comments: 0

$$\:\:\:\:\underbrace{\:} \\ $$

Question Number 210807 Answers: 0 Comments: 0

Question Number 210702 Answers: 2 Comments: 0

How many real solutions does the equation x=sin3x have?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{does}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{x}=\mathrm{sin3}{x}\:\mathrm{have}? \\ $$

Question Number 210248 Answers: 1 Comments: 0

Question Number 209924 Answers: 0 Comments: 0

If f(x)=(x!)∙(x!!)∙(x!!!) find (d/dx)(f(x))=?

$$\boldsymbol{{If}}\:\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\left(\boldsymbol{{x}}!\right)\centerdot\left(\boldsymbol{{x}}!!\right)\centerdot\left(\boldsymbol{{x}}!!!\right)\:\: \\ $$$$\boldsymbol{{find}}\:\:\frac{\boldsymbol{{d}}}{\boldsymbol{{dx}}}\left(\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\right)=? \\ $$

Question Number 209624 Answers: 1 Comments: 0

I=∫_0 ^( ∞) ∫_0 ^( ∞) (( 1)/(1+ x^2 +y^2 +x^2 y^2 )) dxdy=? using polar system...

$$ \\ $$$$ \\ $$$$\mathrm{I}=\int_{\mathrm{0}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \:\frac{\:\mathrm{1}}{\mathrm{1}+\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{x}^{\mathrm{2}} {y}^{\mathrm{2}} }\:{dxdy}=? \\ $$$$\:{using}\:\:\:\:{polar}\:\:{system}... \\ $$

Question Number 209308 Answers: 1 Comments: 0

Donner l′e^ quivalence simple de I_n =∫^( 1) _( 0) (t^n /(t^n −t+1))dt

$$\mathrm{Donner}\:\mathrm{l}'\acute {\mathrm{e}quivalence}\:\mathrm{simple} \\ $$$$\mathrm{de}\:\mathrm{I}_{\mathrm{n}} =\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{{t}^{{n}} }{{t}^{{n}} −{t}+\mathrm{1}}{dt} \\ $$

Question Number 209232 Answers: 1 Comments: 0

u_0 = a, u_(n+1) = (√(u_n v_n )) v_0 = b ∈ ]0,1[ , v_(n+1) = (1/(2(u_n +v_n ))) • show that a≤u_n ≤u_(n+1) ≤v_n ≤v_(n+1) ≤b • show that v_n − u_n ≤ ((a+b)/2^n )

$${u}_{\mathrm{0}} \:=\:{a},\:{u}_{{n}+\mathrm{1}} \:=\:\sqrt{{u}_{{n}} {v}_{{n}} } \\ $$$$\left.{v}_{\mathrm{0}} \:=\:{b}\:\in\:\right]\mathrm{0},\mathrm{1}\left[\:,\:{v}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\left({u}_{{n}} +{v}_{{n}} \right)}\right. \\ $$$$\bullet\:{show}\:{that}\:{a}\leqslant{u}_{{n}} \leqslant{u}_{{n}+\mathrm{1}} \leqslant{v}_{{n}} \leqslant{v}_{{n}+\mathrm{1}} \leqslant{b} \\ $$$$\bullet\:{show}\:{that}\:{v}_{{n}} \:−\:{u}_{{n}} \:\leqslant\:\frac{{a}+{b}}{\mathrm{2}^{{n}} } \\ $$

Question Number 209229 Answers: 6 Comments: 2

Question Number 209228 Answers: 1 Comments: 0

Question Number 209041 Answers: 1 Comments: 2

Question Number 208976 Answers: 4 Comments: 0

Question Number 208855 Answers: 1 Comments: 0

Question Number 208553 Answers: 3 Comments: 0

$$\:\:\:\cancel{ } \\ $$

Question Number 208499 Answers: 0 Comments: 0

Question Number 207954 Answers: 2 Comments: 0

$$\:\underline{\boldsymbol{{x}}} \\ $$

Question Number 207175 Answers: 2 Comments: 0

If x^m .y^n = (x + y)^(m + n) then (d^2 y/dx^2 ) = ?

$$\mathrm{If}\:{x}^{{m}} .{y}^{{n}} \:=\:\left({x}\:+\:{y}\right)^{{m}\:+\:{n}} \:\mathrm{then}\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:=\:? \\ $$

Question Number 207122 Answers: 1 Comments: 0

If e^y (x + 1) = 1 then prove that (d^2 y/dx^2 ) = ((dy/dx))^2 .

$$\mathrm{If}\:{e}^{{y}} \left({x}\:+\:\mathrm{1}\right)\:=\:\mathrm{1}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:=\:\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} . \\ $$

Question Number 207109 Answers: 3 Comments: 0

If y = (1 + x)(1 + x^2 )(1 + x^4 ) .... (1 + x^(2n) ) then find (dy/dx) at x = 0.

$$\mathrm{If}\:{y}\:=\:\left(\mathrm{1}\:+\:{x}\right)\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)\left(\mathrm{1}\:+\:{x}^{\mathrm{4}} \right)\:....\:\left(\mathrm{1}\:+\:{x}^{\mathrm{2}{n}} \right) \\ $$$$\mathrm{then}\:\mathrm{find}\:\frac{{dy}}{{dx}}\:\mathrm{at}\:{x}\:=\:\mathrm{0}. \\ $$

Question Number 206882 Answers: 1 Comments: 0

f(x)=tan^2 x (√(tan x((tan x((tan x((tan x(√(...))))^(1/5) ))^(1/4) ))^(1/3) )) f ′((π/4))=?

$$\:\:{f}\left({x}\right)=\mathrm{tan}\:^{\mathrm{2}} {x}\:\sqrt{\mathrm{tan}\:{x}\sqrt[{\mathrm{3}}]{\mathrm{tan}\:{x}\sqrt[{\mathrm{4}}]{\mathrm{tan}\:{x}\sqrt[{\mathrm{5}}]{\mathrm{tan}\:{x}\sqrt{...}}}}} \\ $$$$\:{f}\:'\left(\frac{\pi}{\mathrm{4}}\right)=? \\ $$

Question Number 206340 Answers: 2 Comments: 0

∫_0 ^( 1) (( ln(1−x )ln(1+x ))/x)dx = Σ_(n=1) ^∞ Ω_n find : Σ_(n=1) ^∞ n Ω_n = ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\:{ln}\left(\mathrm{1}−{x}\:\right){ln}\left(\mathrm{1}+{x}\:\right)}{{x}}{dx}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\Omega_{{n}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{find}\::\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:{n}\:\Omega_{{n}} \:=\:? \\ $$

Question Number 206244 Answers: 1 Comments: 0

$$\:\:\:\zeta \\ $$

Question Number 205827 Answers: 2 Comments: 0

x^3 +y^3 =1 find the implceat second derivative

$$\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} =\mathrm{1} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{implceat}\:\mathrm{second}\:\mathrm{derivative} \\ $$

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