### Home > APCALC > Chapter 7 > Lesson 7.1.1 > Problem7-6

7-6.

Differentiate each equation with respect to $x$. Leave your answers in terms of only $y$ and $x$.

1. $y = 7\ln(x + 1) - x^2$

$\text{The derivative of ln}(x)=\frac{1}{x}.$

1. $2^x + 2^y = e$

Implicit Differentiation.

1. $y = e^{\tan(x)}$

Chain Rule.

1. $(5x + 1)^2 + (y + 1)^2 = 1$

$\text{Implicit differentiation. Solve for }\frac{dy}{dx}.$

1. What is $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for the function in part (a)? Write your answer in terms of $x$ only.

$\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\text{(answer to part (a)) }.$

2. Evaluate $\frac { d ^ { 2 } y } { d x ^ { 2 } } | x = 3$ for the function in part (a).

Translation: Evaluate the 2nd-derivative (that you found in part (e)) at $x = 3$.

3. Write the equation of the tangent line at $x = 3$ for function in part (a).

The equation for the tangent line is $y − y(3) = y^\prime(3)(x − 3)$.

4. If the tangent line is used to approximate the function at $x = 3.01$, will it give an underestimate or an overestimate? Use part (f) to determine your answer.

Is the graph concave up or concave down?