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DifferentiationQuestion and Answers: Page 9

Question Number 165641 Answers: 2 Comments: 0

Given that y = (1/x) (a) Show that y^((n)) = (((−1)^n n!)/x^(n+1) ) (b) Find an expression for y^((n−1)) + y^((n))

$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Given}\:\mathrm{that}\:\:{y}\:=\:\frac{\mathrm{1}}{{x}}\: \\ $$$$\left({a}\right)\:\mathrm{Show}\:\mathrm{that}\:\:{y}^{\left({n}\right)} \:=\:\frac{\left(−\mathrm{1}\right)^{{n}} \:{n}!}{{x}^{{n}+\mathrm{1}} } \\ $$$$\left({b}\right)\:\mathrm{Find}\:\mathrm{an}\:\mathrm{expression}\:\mathrm{for}\:{y}^{\left({n}−\mathrm{1}\right)} +\:{y}^{\left({n}\right)} \\ $$$$ \\ $$

Question Number 165599 Answers: 1 Comments: 0

Question Number 165597 Answers: 0 Comments: 0

prove that Σ_(n=1) ^∞ (( ψ^( (1)) (n))/n^( 2) ) =(7/4) ζ (4) ■ m.n

$$ \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\psi^{\:\left(\mathrm{1}\right)} \left({n}\right)}{{n}^{\:\mathrm{2}} }\:=\frac{\mathrm{7}}{\mathrm{4}}\:\zeta\:\left(\mathrm{4}\right)\:\:\:\blacksquare\:{m}.{n} \\ $$$$ \\ $$

Question Number 165581 Answers: 2 Comments: 0

ϕ(t)=∫_0 ^( (π/2)) ( sin(x)+t cos(x))^( 2) dx find the value of the extermum of ϕ (t).

$$ \\ $$$$\varphi\left({t}\right)=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\:{sin}\left({x}\right)+{t}\:{cos}\left({x}\right)\right)^{\:\mathrm{2}} {dx} \\ $$$${find}\:\:{the}\:\:{value}\:{of}\:{the}\:{extermum} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{of}\:\:\:\varphi\:\left({t}\right). \\ $$

Question Number 165471 Answers: 0 Comments: 1

{ ((h(3x)=(((2−x)/(x+1))−f(x^3 ))^2 )),((f(1)=f ′(1)=2)) :} h ′(3)=?

$$\:\begin{cases}{{h}\left(\mathrm{3}{x}\right)=\left(\frac{\mathrm{2}−{x}}{{x}+\mathrm{1}}−{f}\left({x}^{\mathrm{3}} \right)\right)^{\mathrm{2}} }\\{{f}\left(\mathrm{1}\right)={f}\:'\left(\mathrm{1}\right)=\mathrm{2}}\end{cases} \\ $$$$\:{h}\:'\left(\mathrm{3}\right)=? \\ $$

Question Number 165441 Answers: 1 Comments: 0

Question Number 165328 Answers: 3 Comments: 0

prove that Nice Integral 𝛗=∫_0 ^( 1) (( tan^( −1) (x^( (3/2)) ))/x^( 2) ) dx =((π + (√3) ln(7 +4(√3) ))/4) ■ m.n −−−−−−−−−

$$ \\ $$$$\:\:\:\:\:{prove}\:{that} \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\mathscr{N}{ice}\:\:\:\mathscr{I}{ntegral} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tan}^{\:−\mathrm{1}} \:\left({x}^{\:\frac{\mathrm{3}}{\mathrm{2}}} \right)}{{x}^{\:\mathrm{2}} }\:{dx}\:\:=\frac{\pi\:+\:\sqrt{\mathrm{3}}\:{ln}\left(\mathrm{7}\:+\mathrm{4}\sqrt{\mathrm{3}}\:\right)}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:{m}.{n} \\ $$$$\:\:\:\:\:\:−−−−−−−−−\:\:\: \\ $$

Question Number 165194 Answers: 2 Comments: 0

∫_0 ^( 2π) ln ( 1+ cos (x)).cos (nx )dx=?

$$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} {ln}\:\left(\:\mathrm{1}+\:{cos}\:\left({x}\right)\right).{cos}\:\left({nx}\:\right){dx}=? \\ $$

Question Number 165168 Answers: 1 Comments: 0

Question Number 165152 Answers: 1 Comments: 0

prove Ω=∫_0 ^( 1) (( x − x^( 2) )/((1+x )ln(x))) dx = ln((4/π) ) −−−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{prove} \\ $$$$\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{x}\:−\:{x}^{\:\mathrm{2}} }{\left(\mathrm{1}+{x}\:\right){ln}\left({x}\right)}\:{dx}\:=\:{ln}\left(\frac{\mathrm{4}}{\pi}\:\right) \\ $$$$\:\:\:−−−−− \\ $$

Question Number 165018 Answers: 0 Comments: 0

y = Γ(m+n) Find (dy/dn)

$${y}\:=\:\Gamma\left({m}+{n}\right)\: \\ $$$${Find}\:\frac{{dy}}{{dn}} \\ $$

Question Number 164747 Answers: 1 Comments: 0

faind (dy/dx) sin^(−1) (xy)=csc^(−1) (x−y)

$${faind}\:\:\frac{{dy}}{{dx}} \\ $$$${sin}^{−\mathrm{1}} \left({xy}\right)={csc}^{−\mathrm{1}} \left({x}−{y}\right) \\ $$

Question Number 164716 Answers: 1 Comments: 0

find minimum value of f(x)=4sin 2x−5sin x−5cos x+6

$$\:\:\:{find}\:{minimum}\:{value}\:{of}\: \\ $$$$\:{f}\left({x}\right)=\mathrm{4sin}\:\mathrm{2}{x}−\mathrm{5sin}\:{x}−\mathrm{5cos}\:{x}+\mathrm{6} \\ $$

Question Number 164671 Answers: 1 Comments: 1

solve cos^( 3) (x) + sin^( 2) (x) = (7/8) adopted from youtube ...

$$ \\ $$$$\:\:\:\:\:\:\:\:{solve}\: \\ $$$$\:\:\:\:\:\:{cos}^{\:\mathrm{3}} \left({x}\right)\:+\:{sin}^{\:\mathrm{2}} \left({x}\right)\:=\:\frac{\mathrm{7}}{\mathrm{8}}\: \\ $$$$\:\:\:\:\:\:\:\:\:{adopted}\:{from}\:{youtube}\:... \\ $$$$ \\ $$

Question Number 164653 Answers: 2 Comments: 0

solve 𝛗 = ∫_0 ^( 1) ((ln^( 2) ( x ). tanh^( −1) ( x ))/x)dx =? Ω= ∫_0 ^( 1) (( (tanh^(−1) (x))^( 2) )/(1+x)) = ? −−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:{solve} \\ $$$$\:\:\boldsymbol{\phi}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}^{\:\mathrm{2}} \left(\:{x}\:\right).\:{tanh}^{\:−\mathrm{1}} \left(\:{x}\:\:\right)}{{x}}{dx}\:=? \\ $$$$\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\left({tanh}^{−\mathrm{1}} \left({x}\right)\right)^{\:\mathrm{2}} }{\mathrm{1}+{x}}\:=\:? \\ $$$$\:\:\:\:\:\:−−−− \\ $$

Question Number 164547 Answers: 1 Comments: 0

prove Ω= ∫_0 ^( ∞) (( (√x))/(( 1+x +x^( 2) )^( 3) )) dx =^? ((π(√3))/(36)) −−m.n−−

$$ \\ $$$$\:\:\:\:\:\:\:\:{prove} \\ $$$$\: \\ $$$$\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\sqrt{{x}}}{\left(\:\mathrm{1}+{x}\:+{x}^{\:\mathrm{2}} \right)^{\:\mathrm{3}} \:}\:{dx}\:\overset{?} {=}\:\frac{\pi\sqrt{\mathrm{3}}}{\mathrm{36}}\: \\ $$$$\:\:\:\:\:\:−−{m}.{n}−−\: \\ $$$$ \\ $$

Question Number 164462 Answers: 2 Comments: 2

Find x, such that f(x) is minimum. f(x)={((√(c^2 −x^2 ))/(c−x))−(c−x)}^2

$${Find}\:{x},\:{such}\:{that}\:{f}\left({x}\right)\:{is}\:{minimum}. \\ $$$${f}\left({x}\right)=\left\{\frac{\sqrt{{c}^{\mathrm{2}} −{x}^{\mathrm{2}} }}{{c}−{x}}−\left({c}−{x}\right)\right\}^{\mathrm{2}} \\ $$

Question Number 164366 Answers: 1 Comments: 1

(d/dx) (e^(tan(x)) ) {Z.A}

$$\frac{\boldsymbol{{d}}}{\boldsymbol{{dx}}}\:\left(\boldsymbol{{e}}^{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)} \right) \\ $$$$\left\{\boldsymbol{{Z}}.\boldsymbol{\mathrm{A}}\right\} \\ $$

Question Number 164339 Answers: 1 Comments: 0

In AB^Δ C : cos^( 2) (A )+ cos^( 2) (B )+ cos^( 2) ( C )=1 . Prove that AB^Δ C is right angled. −−−−−−−−

$$ \\ $$$$\:\:\:\:{In}\:\:{A}\overset{\Delta} {{B}C}\:\:\::\:\:\:{cos}^{\:\mathrm{2}} \left({A}\:\right)+\:{cos}^{\:\mathrm{2}} \left({B}\:\right)+\:{cos}^{\:\mathrm{2}} \left(\:{C}\:\right)=\mathrm{1}\:\:. \\ $$$$\:\:\:\:\:\:\:\:{Prove}\:{that}\:\:{A}\overset{\Delta} {{B}C}\:\:\:{is}\:\:\:{right}\:{angled}. \\ $$$$\:\:\:\:\:\:−−−−−−−− \\ $$$$\:\:\:\:\:\: \\ $$

Question Number 164103 Answers: 1 Comments: 0

prove that Ω=∫_0 ^( 1) ln(((1+x)/(1−x)) ).(dx/(x (√( 1−x^( 2) )))) = (π^( 2) /2) −− m.n−−

$$ \\ $$$$\:\:\:{prove}\:{that} \\ $$$$\: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\:\right).\frac{{dx}}{{x}\:\sqrt{\:\mathrm{1}−{x}^{\:\mathrm{2}} }}\:=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{2}} \\ $$$$\:\:\:\:\:−−\:{m}.{n}−− \\ $$$$ \\ $$

Question Number 164059 Answers: 0 Comments: 0

Air leaks from a spherical ballon so that it maintains its shape at a rate of 25 cc/m .What is the rate of change in the length of the radius of the balloon when the radius is 5 cm

$$\:\:\mathrm{Air}\:\mathrm{leaks}\:\mathrm{from}\:\mathrm{a}\:\mathrm{spherical}\:\mathrm{ballon}\:\mathrm{so}\:\mathrm{that}\: \\ $$$$\:\mathrm{it}\:\mathrm{maintains}\:\mathrm{its}\:\mathrm{shape}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{25}\:\mathrm{cc}/\mathrm{m} \\ $$$$\:.\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{change}\:\mathrm{in}\:\mathrm{the}\:\mathrm{length} \\ $$$$\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{balloon}\:\mathrm{when}\:\mathrm{the}\:\mathrm{radius} \\ $$$$\:\:\mathrm{is}\:\mathrm{5}\:\mathrm{cm} \\ $$

Question Number 163928 Answers: 1 Comments: 0

f(x)=((2x^(100!) )/(100!))+x^(100) +1 find ((d^(100!) f(x))/dx^(100!) )=?

$${f}\left({x}\right)=\frac{\mathrm{2}{x}^{\mathrm{100}!} }{\mathrm{100}!}+{x}^{\mathrm{100}} +\mathrm{1} \\ $$$${find}\:\:\:\frac{{d}^{\mathrm{100}!} {f}\left({x}\right)}{{dx}^{\mathrm{100}!} }=? \\ $$

Question Number 163732 Answers: 0 Comments: 0

If , 𝛗 = ∫_(−∞) ^( +∞) (( sin(x).ln^( 2) (x ))/x) then find : Im (𝛗 ) = ? ■ m.n

$$ \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\mathrm{I}{f}\:,\:\:\boldsymbol{\phi}\:=\:\int_{−\infty} ^{\:+\infty} \frac{\:{sin}\left({x}\right).{ln}^{\:\mathrm{2}} \left({x}\:\right)}{{x}}\:\:{then} \\ $$$$\:\:\:\:\:\:{find}\:\::\:\:\:\:\:\:\:\:\:\mathcal{I}{m}\:\left(\boldsymbol{\phi}\:\right)\:=\:?\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\: \\ $$$$ \\ $$

Question Number 163704 Answers: 2 Comments: 0

Ω= Σ_(n=1) ^∞ n(ζ (1+n) −1) =?

$$ \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{n}\left(\zeta\:\left(\mathrm{1}+{n}\right)\:−\mathrm{1}\right)\:=? \\ $$$$ \\ $$

Question Number 163700 Answers: 3 Comments: 0

K(x) = ((3 cos x)/(5+4sin x)) {: ((max K(x))),((min K(x))) } =?

$$\:\:\mathcal{K}\left({x}\right)\:=\:\frac{\mathrm{3}\:\mathrm{cos}\:{x}}{\mathrm{5}+\mathrm{4sin}\:{x}} \\ $$$$\:\left.\begin{matrix}{{max}\:\mathcal{K}\left({x}\right)}\\{{min}\:\mathcal{K}\left({x}\right)}\end{matrix}\right\}\:=? \\ $$

Question Number 163682 Answers: 1 Comments: 0

Ω= ∫_0 ^( 1) (Li_( 2) ^ (x ))^( 2) dx = ? ■ m.n −−− −−−

$$ \\ $$$$\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\left(\mathrm{Li}_{\:\mathrm{2}} ^{\:} \left({x}\:\right)\right)^{\:\mathrm{2}} {dx}\:=\:?\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\: \\ $$$$\:\:\:\:\:−−−\:−−− \\ $$$$\:\:\:\: \\ $$

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