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In Problems 1–42, find the derivative of each function. When a, b, or n appear, they are constants.
\(y=(ax+b)^{n}\)
\(y=\sqrt{2ax}\)
\(y=x\sqrt{1-x}\)
\({y=\frac{1}{\sqrt{x^{2}+1}}}\)
\(y=(x^{2}+4)^{3/2}\)
\(F(x) =\frac{x^{2}}{\sqrt{x^{2}-1}}\)
\(z={\frac{\sqrt{2ax-x^{2}}}{x}}\)
\(y=\sqrt{x}+\sqrt[3]{x}\)
\(y=(e^{x}-x)^{5x}\)
\(\phi (x)={\frac{(x^{2}-a^{2})^{3/2}}{\sqrt{x+a}}}\)
\(f(x)={\frac{x^{2}}{(x-1)^{2}}}\)
\(u= (b^{1/2}-x^{1/2}) ^{2}\)
\(y=x\sec (2x)\)
\(u=\cos ^{3}x\)
\(y=\sqrt{a^{2}\sin \left( \frac{x}{a}\right)}\)
\(\phi (z)=\sqrt{1+\sin z}\)
\(u=\sin v-\frac{1}{3}\sin ^{3}v\)
\(y=\tan \sqrt{\frac{\pi }{x}}\)
\(y= (1.05) ^{x}\)
\(v=\ln (y^{2}+1)\)
\(z=\ln (\sqrt{u^{2}+25} - u)\)
\(y=x^{2}+2^{x}\)
\(y=\ln [\sin (2x)]\)
\(f(x)=e^{-x}\sin (2x+\pi)\)
\(g(x)=\ln (x^{2}-2x)\)
\(y=\ln \frac{x^{2}+1}{x^{2}-1}\)
\(y=e^{-x}\ln x\)
\(w=\ln \!\big(\sqrt{x+7}-\sqrt{x}\big)\)
\(y=\frac{1}{12}\ln \left( \frac{x}{\sqrt{144-x^{2}}}\right)\)
\(y=\ln (\tan ^{2}x)\)
\(f(x) =\frac{e^{x}(x^{2}+4) }{(x-2)}\)
\(y=\sin^{-1} (x-1) + \sqrt{2x-x^{2}}\)
\(y=2\sqrt{x}-2\tan ^{-1}\sqrt{x}\)
\(y=4\tan ^{-1}\frac{x}{2}+x\)
\(y=\sin ^{-1}(2x-1)\)
\(y=x^{2}\tan ^{-1}\frac{1}{x}\)
\(y=x\tan ^{-1}x-\ln \sqrt{1+x^{2}}\)
\(y=\sqrt{1-x^{2}}(\sin ^{-1}x)\)
\(y=\tanh \frac{x}{2}+\frac{2x}{4+x^{2}}\)
\(y=x\sinh x\)
\(y=\sqrt{\sinh x}\)
\(y=\sinh ^{-1}e^{x}\)
In Problems 43–48, find \(y'=\frac{dy}{dx}\) using implicit differentiation.
\(x=y^{5}+y\)
\(x=\cos ^{5}y+\cos y\)
\(\ln x+\ln y=x\cos y\)
\(\tan (xy)=x\)
\(y=x+\sin (xy)\)
\(x=\ln (\csc y+\cot y)\)
In Problems 49–52, find \(y^\prime\) and \(y^{\prime\prime}\).
\(xy+3y^{2}=10x\)
\(y^{3}+y=x^{2}\)
\(xe^{y}=4x^{2}\)
\(\ln (x+y) =8x\)
The function \(f(x) =e^{2x}\) has an inverse function \(g.\) Find \(g^\prime (1)\).
The function \(f(x) =\sin x\) defined on the restricted domain \(\left[ -\frac{\pi }{2},\,\frac{\pi }{2}\right]\) has an inverse function \(g.\) Find \(g^\prime \left(\frac{1}{2}\right)\).
In Problems 55–56, express each limit in terms of the number \(e\).
\(\lim\limits_{n\rightarrow \infty }\left( 1+\frac{2}{5n}\right) ^{n}\)
\(\lim\limits_{h\rightarrow 0} ( 1+3h) ^{2/h}\)
In Problems 57 and 58, find the exact value of each expression.
\(\sinh 0\)
\(\cosh ( \ln 3)\)
In Problems 59 and 60, establish each identity.
\(\sinh x+\cosh x=e^{x}\)
\(\tanh (x+y) =\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}\)
If \(f(x)=\sqrt{1-\sin ^{2}x}\), find the domain of \(f^\prime\).
If \(f(x)=x^{1/2}(x-2)^{3/2}\) for all \(x ≥ 2\), find the domain of \(f^\prime\).
Let \(f\) be the function defined by \(f(x)=\sqrt{1+6x}.\)
Tangent and Normal Lines Find equations of the tangent and normal lines to the graph of \(y=x\sqrt{x+(x-1)^{2}}\) at the point \((2,\,2 \sqrt{3})\).
Find the differential \(dy\) if \(x^{3}+2y^{2}=x^{2}y\).
Measurement Error If \(p\) is the period of a pendulum of length \(L\), the acceleration due to gravity may be computed by the formula \( g=\frac{(4\pi ^{2}L)}{p^{2}}\). If \(L\) is measured with negligible error, but a \(2{\%}\) error may occur in the measurement of \(p\), what is the approximate percentage error in the computation of \(g\)?
Linear Approximation Find a linear approximation to \[ \hbox{\(y=x+\ln x\) at \(x=1\).} \]
Measurement Error If the percentage error in measuring the edge of a cube is \(5{\%}\), what is the percentage error in computing its volume?
For the function \(f(x) =\tan x\):
For the function \(f(x) =\ln x\):
If \(f(x)=(x^{2}+1)^{(2-3x)}\), find \(f^\prime (1)\).
Find \(\lim\limits_{x\rightarrow 2}\frac{\ln x-\ln 2}{x-2}\).
Find \(y^{\prime }\) at \(x=\frac{\pi }{2}\) and \(y=\pi\) if \(x\sin y+y\cos x=0\).
In Problems 74–77, find the Taylor Polynomial \(P_{n}( x)\) for \(f\) at \(x_{0}\) for the given \(n\) and \(x_{0}\).
\(f(x)=e^{2x};\quad n=4, x_{0}=3\)
\(f(x)=\tan x;\quad n=4, x_{0}=0\)
\(f(x)=\frac{1}{1+x}; \quad n=4, x_{0}=1\)
\(f(x)=\ln x; n=6, x_{0}=2\)
In Problems 78 and 79, for each function:
\(f(x)=8x^{4}-2x^{2}+5x-1\), interval: \((0,1)\). Let \(c_{1}=0.5.\)
\(f(x)=2-x+\sin x\), interval: \(\left(\frac{\pi }{2},\pi \right)\). Let \(c_{1}=\frac{\pi}{2}.\)
Tangent Line Find an equation of the tangent line to the graph of \(4xy-y^{2}=3\) at the point \(( 1, 3)\).