Derivatives of inverse functions

For each function given below, find its derivative.

1. $h(x) = x^2\ln(x)$

2. $p(t) = \frac{\ln(t)}{e^t + 1}$

3. $s(y) = \ln(\cos(y) + 2)$

4. $z(x) = \tan(\ln(x))$

5. $m(z) = \ln(\ln(z))$

The following prompts in this activity will lead you to develop the derivative of the inverse tangent function.

1. Let $r(x) = \arctan(x)$. Use the relationship between the arctangent and tangent functions to rewrite this equation using only the tangent function.

2. Differentiate both sides of the equation you found in (a). Solve the resulting equation for $r'(x)$, writing $r'(x)$ as simply as possible in terms of a trigonometric func…

Determine the derivative of each of the following functions.

1. $\displaystyle f(x) = x^3 \arctan(x) + e^x \ln(x)$

2. $\displaystyle p(t) = 2^{t\arcsin(t)}$

3. $\displaystyle h(z) = (\arcsin(5z) + \arctan(4-z))^{27}$

4. $\displaystyle s(y) = \cot(\arctan(y))$

5. $\displaystyle m(v) = \ln(\sin^2(v)+1)$

6. $\displaystyle g(w) = \arctan\left( \frac{\ln(w)}{1+w^2} \right)$

Determine the derivative of each of the following functions. Use proper notation and clearly identify the derivative rules you use.

1. $f(x) = \ln(2\arctan(x) + 3\arcsin(x) + 5)$

2. $r(z) = \arctan(\ln(\arcsin(z)))$

3. $q(t) = \arctan^2(3t) \arcsin^4(7t)$

4. $g(v) = \ln\left( \frac{\arctan(v)}{\arcsin(v) + v^2} \right)$

Let $f(x) = \frac{1}{4}x^3 + 4$.

1. Sketch a graph of $y = f(x)$ and explain why $f$ is an invertible function.

2. Let $g$ be the inverse of $f$ and determine a formula for $g$.

3. Compute $f'(x)$, $g'(x)$, $f'(2)$, and $g'(6)$. What is the special relationship between $f'(2)$ and $g'(6)$? Why?

Let $h(x) = x + \sin(x)$.

1. Sketch a graph of $y = h(x)$ and explain why $h$ must be invertible.

2. Explain why it does not appear to be algebraically possible to determine a formula for $h^{-1}$.

3. Observe that the point $(\frac{\pi}{2}, \frac{\pi}{2} + 1)$ lies on the graph of $y = h(x)$. Determine the value of $(h^{-1})'(\frac{\pi}{2} + 1)$.