Compute the derivative of f(x)=xarcsin⁡x a) xarcsin⁡x(ln⁡x1−x2+arcsin⁡xx) b) xarcsin⁡x(11−x2+arcsin⁡xx) c) 11−x2 d) 11−x2+arcsin⁡xx e) 11−x2+arcsin⁡x

Question

Compute the derivative of f(x)=xarcsin⁡x  a) xarcsin⁡x(ln⁡x1−x2+arcsin⁡xx)  b) xarcsin⁡x(11−x2+arcsin⁡xx)  c) 11−x2  d) 11−x2+arcsin⁡xx  e) 11−x2+arcsin⁡x

braodagxj · Accepted Answer

Find the derivative of the following via implicit differentiation:ddx(f(x))=ddx(xsin−1(x))Using the chain rule, ddx(f(x))=df(u)dududx, where u=x and ddu(f(u))=f′(u):(ddx(x))f′(x)=ddx(xsin−1(x))The derivative of x is 1:1f′(x)=ddx(xsin−1(x))Express xsin−1(x) as a power of e: xsin−1(x)=elog⁡(xsin−1(x))=elog⁡(x)(sin−1(x)):f′(x)=ddx(e(sin−1(x))log⁡(x))Using the chain rule, ddx(elog⁡(x)(sin−1(x)))=deudududx, where u=log⁡(x)(sin−1(x)) and ddu(eu)=eu:f′(x)=(ddx((sin−1(x))log⁡(x)))e(sin−1(x))

Pansdorfp6 · Accepted Answer

Hmmm, I think that is wrong answer.

nick1337 · Accepted Answer

Given that f(x)=xarcsin⁡x=xsin−1⁡xlog⁡f(x)=log⁡xsin−1⁡xddx(log⁡f(x))=ddx(sin−1⁡x⋅log⁡x)1f(x)ddxf(x)=sin−1⁡xddxlog⁡x+log⁡xddx(sin−1⁡x)1f(x)f′(x)=sin−1⁡x1xlog⁡x+log⁡x11−x2f′(x)=f(x)[sin−1⁡xx+log⁡x1−x2]⇒f′(x)=xsin−1⁡x[sin−1⁡xx+log⁡x1−x2][∴d(log⁡x)=1x, d(sin−1⁡x)=11−x2

Compute the derivative of f(x)=x^{\arcsin x} a) x^{\arcsin x}(\frac{\ln x}{\sqrt{1-x^2}}+\frac{\arcsin x}{x}) b)

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