Find \( f'(x) \).
\[ f(x)=\frac{e^{2x}}{2x\cdot e^{x-1}} \]
Find \( f'(x) \).
\[ f(x)=\sin^{-1}{(x)}-x\sqrt{1-x^2} \]
Find \( f''(x) \).
\[ f(x)=-6x^{3/2} \]
Find \( g''(x) \).
\[ g(x)=\frac{3x}{\sqrt{3-x^2}} \]
Find \( g''(x) \).
\[ g(x)=\arctan{x}+\frac{x}{1+x^2} \]
Write an equation for the tangent line to the graph at the given \( x \)-value.
\[ f(x)=\frac{1}{2}x^4-3x+6 \] \[ x=1 \]
Write an equation for the tangent line to the graph at the given \( x \)-value.
\[ f(x)=x^2 \cdot \sin{x} \] \[ x=\frac{\pi}{2} \]
Write an equation for the tangent line to the graph at the given \( x \)-value.
\[ f(x)=\sqrt{x^2+x} \] \[ x=1 \]
Write an equation for the tangent line to the graph at the given \( x \)-value.
\[ g(x)=\sqrt{x}-\frac{1}{4}e^x \] \[ x= \ln{16} \]
Write an equation for the tangent line to the graph at the given \( x \)-value.
\[ f(x)=2x+e^{2x} \] \[ x=0 \]
Write an equation for the tangent line to the graph at the given \( x \)-value.
\[ f(x)=x^4-4x^3+5x+3 \] \[ x=1 \]
Find \( k \) such that the line is tangent to the graph of the function.\[ f(x)=kx^{2/3} \] \[ y=-2x-8 \]
Find equations of the tangent lines to the graph of the function that are parallel to the given line, \( 8x-2y=-13 \).\[ f(x)=\frac{x-2}{x+2} \]
Show that the graph of the function does not have a horizontal tangent line. \[ f(x) =5x+\cos{x}-4 \]
Determine \( \frac{dy}{dx} \) by implicit differentiation.
\[ x^2y^2-4x^2-9y^2=0 \]
Determine \( \frac{dy}{dx} \) by implicit differentiation.
\[ 4x^3 + \ln{y^2}+y=4x \]
Determine \( \frac{dy}{dx} \) by implicit differentiation.
\[ -\cos{(x-y)}=x+y \]
Determine \( \frac{d^2y}{dx^2} \) in terms of \( x \) and \( y. \)
\[ x^2+y^2=81 \]
Determine \( \frac{d^2y}{dx^2} \) in terms of \( x \) and \( y. \)
\[ x^2y+4x=8 \]
Determine \( \frac{d^2y}{dx^2} \) in terms of \( x \) and \( y. \)
\[ xy-x^2=-6 \]
Write an equation of the tangent line to the graph of the equation at the given point.
\[ xy=10 \quad (-5, -2) \]
Write an equation of the tangent line to the graph of the equation at the given point.
\[ e^{xy}-2x=0 \quad (1, \ln{2}) \]
Write an equation of the tangent line to the graph of the equation at the given point.
\[ x^2+xy+y^2=9 \quad (-3, 0) \]
Write an equation of the tangent line to the graph of the equation at the given point.
\[ y^2 =\ln{xy} \quad (e, 1) \]
Derive the Product Rule
Using Logarithmic Differentiation\[ y=f(x) \cdot g(x) \]
Derive the Quotient Rule
Using Logarithmic Differentiation\[ y=\frac{f(x)}{g(x)} \]
Find \( \frac{dy}{dx} \) using logarithmic differentiation.
\[ y=x^{\sqrt{x}} \]
Find \( \frac{dy}{dx} \) using logarithmic differentiation.
\[ y=x^{\ln{x}} \]
Find \( \frac{dy}{dx} \) using logarithmic differentiation.
\[ y=(\ln{x})^{\ln{x}} \]
Find \( \frac{dy}{dx} \) using logarithmic differentiation.
\[ y=(\ln{x})^{x} \]
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