Exam 33: Derivatives of Trigonometric, Logarithmic, and Exponential Functions

Use Newton's method to find the smallest positive root of the equation $y=(\sin x) \ln (\sin x)$ .

Find the slope of the tangent line at the given value of $x . y=\cos 2 x$ at $x=2$ rad, to two decimal places.

Use Newton's method to find a root of $y{=x+\tan x}$ on [4,5], to three decimal places.

Differentiate: $y=x^{x}$

Differentiate: $y=4 b^{5 x}$

Find the equations of the tangents to the curve $y=\tan^{-1} x$ having a slope of $\frac{1}{2}$ .

Determine the derivative: $y=\log _{a}(x-1)^{3}$

The acceleration of a block of mass $m$ being pulled by a constant force $F$ which is applied at an angle above the horizontal is given by $a=\frac{F \cos \theta-f m g+f F \sin \theta}{m}$ where $f$ is the coefficient of friction between the block and the surface and $g$ is the acceleration due to gravity. Find the angle that maximizes the block's acceleration when $f=0.52$ .

Differentiate: $y=x \log \sqrt{10 x+1}$

Find the derivative: $y=\sin 2 x \cos 2 x$

Find the derivative: $y=5 \tan 2 \pi x-3 \cos 4 \pi x$

Differentiate: $y=\sin 3 x \sec 5 x$

Find the derivative of $y=\frac{2 \sin 3 x}{3 \cot 2 x}$ .

Find the slope of the tangent to the curve $y=x^{2} \sin x^{2}$ at $x=0.5$ .

Differentiate: $y=(\sin x)^{X}$

Find the derivative of $y=x \sec^{-1} x$ .

Find the derivative of $y=\csc x^{2}$ .

Find the derivative of $y=3 \ln \frac{2}{x}$ .

Find the second derivative of $y=\sin^{2} x$ .

Use Newton's method to find a root of $y=x \sin x^{2}$ on [2,3], to three decimal places.