  ### Home > CALC > Chapter 7 > Lesson 7.1.3 > Problem7-31

7-31.

Consider the curve $xe^{5y} = 3y$.

1. Find $\frac { d y } { d x }$.

Implicitly differentiate.

2. Find the equation of the tangent line at $\left(0, 0\right)$.

Notice that the derivative you found in part (a) has both $x$- and $y$-values.

3. If $x = 0.1$, estimate $y$ using the tangent line.

Since $x = 0.1$ is easier to evaluate in a linear equation than in the curve shown above, use the tangent line to estimate the $y$-value at $x = 0.1$.

4. Using $\frac { d ^ { 2 } y } { d x ^ { 2 } }$, determine if the tangent approximation is an over or under estimate. Justify your answer in words.

When finding $\frac{d^{2}y}{dx^{2}}$, remember to substitute your answer in part (a).

The sign of the 2nd derivative at $(0,0)$ will determine if the tangent line is above or below the curve.

When a function is concave up, the tangent line will be below the curve.
When it is concave down, the tangent line will be above the curve.
Verify these statements by sketching a few examples.