UpStudy Free Solution:

To find the slope of the curve \(y^ 5 + x^ 3 = y^ 2 + 12x\) at the point \(( 0, 1) \), we need to find the derivative of \(y\) with respect to \(x\) using implicit differentiation.

Given:

\[y^ 5 + x^ 3 = y^ 2 + 12x\]

Differentiate both sides with respect to \(x\):

\[\frac { d} { dx} ( y^ 5) + \frac { d} { dx} ( x^ 3) = \frac { d} { dx} ( y^ 2) + \frac { d} { dx} ( 12x) \]

Using the chain rule, we get:

\[5y^ 4 \frac { dy} { dx} + 3x^ 2 = 2y \frac { dy} { dx} + 12\]

Rearrange the terms to solve for \(\frac { dy} { dx} \):

\[5y^ 4 \frac { dy} { dx} - 2y \frac { dy} { dx} = 12 - 3x^ 2\]

Factor out \(\frac { dy} { dx} \):

\[( 5y^ 4 - 2y) \frac { dy} { dx} = 12 - 3x^ 2\]

Solve for \(\frac { dy} { dx} \):

\[\frac { dy} { dx} = \frac { 12 - 3x^ 2} { 5y^ 4 - 2y} \]

Now, substitute the point \(( 0, 1) \) into the equation:

\[\frac { dy} { dx} \bigg | _ { ( 0, 1) } = \frac { 12 - 3( 0) ^ 2} { 5( 1) ^ 4 - 2( 1) } = \frac { 12 - 0} { 5 - 2} = \frac { 12} { 3} = 4\]

So, the slope of the curve at the point \(( 0, 1) \) is \(4\).

Therefore, the correct answer is:

b. 4

Supplemental Knowledge

When dealing with curves defined implicitly by an equation involving both \(x\) and \(y\), we use implicit differentiation to find the derivative. This allows us to determine the slope of the tangent line at a given point.

Implicit differentiation involves differentiating both sides of the equation with respect to \(x\), treating \(y\) as a function of \(x\). This means applying the chain rule to terms involving \(y\).

For an equation \(F( x, y) = 0\), we differentiate both sides:

\[\frac { d} { dx} F( x, y) = 0\]

This often results in an expression involving both \(x\), \(y\), and \(\frac { dy} { dx} \). We then solve for \(\frac { dy} { dx} \) to find the slope of the curve at a specific point.

UpStudy is your premier platform for tackling complex problems in mathematics, physics, chemistry, and biology. UpStudy Calculus solver delivers accurate, step-by-step solutions instantly. Plus, our expert tutors are available online around the clock to provide personalized support whenever you need it. With UpStudy by your side, mastering challenging subjects has never been easier!