Problems

01

Arctrig derivatives

Write down a complete table of formulas for $\frac{d}{dx}{\sin }^{-1}x$, $\frac{d}{dx}{\cos }^{-1}x$, $\frac{d}{dx}{\tan }^{-1}x$, $\frac{d}{dx}{\csc }^{-1}x$, $\frac{d}{dx}{\sec }^{-1}x$, $\frac{d}{dx}{\cot }^{-1}x$.

Rewrite the above complete table two more times. (Your submission should include at least 3 handwritten copies of the same table.)

Now compute the following derivatives by applying the formulas above in combination with derivative rules like the chain rule:

• (a) $y={\sin }^{-1}\left(3x^2\right)$
• (b) $y=\arctan \left(\sqrt{\frac{1-x}{1+x}}\right)$

02

Cars move away perpendicularly

Two cars leave the same origin, one travels north at $48mi/hr$ and the other west at $20mi/hr$. After two hours, how fast is their absolute distance increasing?

(Hint: write down the distance equation and derive both sides with respect to time.)

03

Implicit differentiation - second derivative $y^{\prime \prime }$

Find $y^{\prime \prime }(x)$ given the relationship $x^2+4y^2=4$.

04

Implicit differentiation practice

Use implicit differentiation to find $y^{\prime }=\frac{dy}{dx}$ when given each of these relationships:

• (a) $y\cos x=x^2+y^2$
• (b) $x\sin y+y\sin x=1$

05

Derivative of $\ln x$

Compute the derivative of the following functions:

• (a) $y=\ln \left(e^{-x}-x{e}^{-x}\right)$
• (b) $y=\ln \left(x^2-3x+1\right)$

Now:

• (c) Find the tangent line to the curve in (b) at $x=3$.

06

Logarithmic differentiation

Using logarithmic differentiation, find the derivatives of the following functions:

• (a) $h(x)=x^{1/x}$
• (b) $p(x)=x^{\sin x}$

Logarithmic differentiation is the only method for deriving $f(x{)}^{g(x)}$, but it also simplifies the process for deriving complicated products and quotients of powers.

Using logarithmic differentiation, find the derivatives of the following functions:

• (c) $f(x)=(x^2+2{)}^2(x^4+4{)}^4$
• (d) $y=\frac{x^{3/4}\sqrt{x^2+1}}{(3x+2{)}^5}$

07

Accumulating sand

Sand is accumulating in a conical pile which has radius always $3\times$ its height. Suppose the sand is falling at a rate of $120\mathrm{f}{\mathrm{t}}^3/min$. How quickly is the height of the pile increasing when the height is $10\mathrm{ft}$?