Home > CALC > Chapter 7 > Lesson 7.1.1 > Problem7-6

7-6.

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

1. $y = 7\operatorname{ln}(x + 1) − x^2$

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

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

Implicit Differentiation.

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

Chain Rule.

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

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

1. Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for part (a). Be sure to leave all parts in terms of y and x only.

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

1. Find $\frac { d ^ { 2 } y } { d x ^ { 2 } } | x = 3$ for part (a).

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

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

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

1. Is the tangent line an over or under approximation? Use part (f) to determine your answer.

Is the graph concave up or concave down?