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7-6.

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

$y = 7\operatorname{ln}(x + 1) − x^2$
Hint (a):
The derivative of $\operatorname{ln}(x)=\frac{1}{x}$ .

$2^x + 2^y = e$
Hint (b):
Implicit Differentiation.

$y = e^{\operatorname{tan}(x)}$
Hint (c):
Chain Rule.

$(5x + 1)^2 + (y + 1)^2 = 1$
Hint (d):
Implicit differentiation. Solve for $\frac{dy}{dx}$ .

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

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } } | x = 3$ for part (a).
Hint (f):
Translation: Evaluate the 2nd-derivative (that you found in part (e)) at $x = 3$ .

Write the equation of the tangent line at $x = 3$ for part (a).
Hint (g):
The equation for the tangent line is $y − y(3) = y^\prime(3)(x − 3)$ .

Is the tangent line an over or under approximation? Use part (f) to determine your answer.
Hint (h):
Is the graph concave up or concave down?

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