Chapter 29

Find the derivative. $$y=t^{2} \operatorname{Arcsin} \frac{t}{2}$$

Integrate $$\int \sec (2 \theta+3) d \theta$$

Exponential Functions $$\int_{2}^{3} x e^{-x^{2}} d x$$

Derivative of \(e^{u}\). Differentiate. $$y=x e^{-x}$$

First Derivatives Find the derivative. $$y=\sin 2 x \cos 3 x$$

Find the derivative. $$y=\sin ^{-1} \frac{x}{\sqrt{1+x^{2}}}$$

Find the derivative. $$z=2 \sin 2 \theta \cot 8 \theta$$

Derivative of In \(u\) Differentiate. $$s=\ln \sqrt{t-5}$$

Exponential Functions $$\int \frac{e^{\sqrt{x-2}}}{\sqrt{x-2}} d x$$

First Derivatives Find the derivative. $$y=1.23 \sin ^{2} x \cos 3 x$$

Derivative of In \(u\) Differentiate. $$y=5.06 \ln \sqrt{x^{2}-3.25 x}$$

Integrate $$\int 3 x^{2} \cot \left(8 x^{3}+3\right) d x$$

Exponential Functions $$\int \frac{\left(e^{x / 2}-e^{-x / 2}\right)^{2}}{4} d x$$

Derivative of \(e^{u}\). Differentiate. $$y=\frac{x^{2}-2}{e^{3 x}}$$

First Derivatives Find the derivative. $$y=\frac{1}{2} \sin ^{2} x$$

Find the slope of the tangent to each curve. $$y=x \operatorname{Arcsin} x \quad \text { at } \quad x=\frac{1}{2}$$

With Trigonometric Functions Differentiate. $$y=\ln \sin x$$

Integrate $$\int x \cos x^{2} d x$$

Exponential Functions $$\int\left(e^{x / a}+e^{-x / a}\right) d x$$

Derivative of \(e^{u}\). Differentiate. $$y=\frac{e^{x}-e^{-x}}{x^{2}}$$

First Derivatives Find the derivative. $$y=\sqrt{\cos 2 t}$$

Find the slope of the tangent to each curve. \(y=\frac{\text { Arctan } x}{x}\) at \(x=1\)

With Trigonometric Functions Differentiate. $$y=\ln \sec x$$

Integrate $$\int 3 x^{2} \cos x^{3} d x$$

Derivative of \(e^{u}\). Differentiate. $$y=\frac{e^{x}-x}{e^{-x}+x^{2}}$$

Exponential Functions $$\int\left(e^{x / a}-e^{-x / a}\right)^{2} d x$$

First Derivatives Find the derivative. $$y=42.7 \sin ^{2} t$$

With Trigonometric Functions Differentiate. $$y=\sin x \ln \sin x$$

Second Derivatives For problems 21 through \(23,\) find the second derivative of each function. $$y=\cos x$$

Integrate $$\int_{0}^{\pi} \sin \phi d \phi$$

Derivative of \(e^{u}\). Differentiate. $$y=\left(x+e^{x}\right)^{2}$$

Find the slope of the tangent to each curve. $$y=\sqrt{x} \operatorname{Arccot} \frac{x}{4} \text { at } x=4$$

Find the second derivative. $$y=3 \tan x$$

With Trigonometric Functions Differentiate. $$y=\ln (\sec x+\tan x)$$

Second Derivatives For problems 21 through \(23,\) find the second derivative of each function. $$y=\frac{1}{4} \cos 2 \theta$$

Integrate $$\int_{0}^{\pi / 2} \cos \phi d \phi$$

Derivative of \(e^{u}\). Differentiate. $$y=\left(e^{x}+2 x\right)^{3}$$

Find the second derivative. $$y=2 \sec 5 \theta$$

Implicit Relations Find \(d y / d x\) $$y \ln y+\cos x=0$$

Second Derivatives For problems 21 through \(23,\) find the second derivative of each function. $$y=x \cos x$$

Integrate $$\int_{0}^{\pi} \cos \frac{\theta}{2} d \theta$$

Logarithmic Functions $$\int_{1}^{2} x \ln x^{2} d x$$

Derivative of \(e^{u}\). Differentiate. $$y=\frac{\left(1+e^{x}\right)^{2}}{x}$$

Implicit Relations Find \(d y / d x\) $$\ln x^{2}-2 x \sin y=0$$

Second Derivatives For problems 21 through \(23,\) find the second derivative of each function. If \(f(x)=x^{2} \cos ^{3} x,\) find \(f^{\prime \prime}(0)\)

Integrate $$\int_{\pi / 3}^{\pi / 2} \sin ^{2} x \cos x d x$$

Derivative of \(e^{u}\). Differentiate. $$y=\left(\frac{e^{x}+1}{e^{x}-1}\right)^{2}$$

Implicit Relations Find \(d y / d x\) $$x-y=\ln (x+y)$$

Second Derivatives For problems 21 through \(23,\) find the second derivative of each function. If \(f(x)=x \sin (\pi / 2) x,\) find \(f^{\prime \prime}(1)\)

$$y=\sin x \quad \text { from } x=0 \text { to } \pi$$