Differential Calculus

To find the equation of the tangent to the curve given by at the point (1, 4), we will use implicit differentiation. Differentiate both sides with respect to .

Using the chain rule and power rule, we have:

Solve for to find the slope of the tangent:

Now substitute and into the derivative to find the slope at (1, 4):

With the slope at (1, 4), use the point-slope form of the line equation :

Simplify this equation:

Multiply through by 8 to clear fractions:

Rearrange to form the equation:

The equation of the tangent to the curve at point (1, 4) is .

We start with the given equation:

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Take the natural logarithm of both sides:

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Simplify using logarithmic properties:

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Differentiate with respect to :

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Differentiate again to find the second derivative:

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Substitute and into the given options. For option (4):

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Simplify:

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Thus, option (4) satisfies the equation.