Please help me with solving these. I’d really appreciate your help. Thank you very much.

Problem 17)

Problem 18)

Step-by-step explanation:

Problem 17)

We have the curve represented by the equation:

And we want to find the equation of the tangent line to the point (1, 1).

First, let's find the derivative dy/dx. Take the derivative of both sides with respect to x:

Simplify. Recall that the derivative of a constant is zero.

Differentiate. We can differentiate the first term normally. The second term will require the product rule. Hence:

Rewrite:

Therefore:

So, the slope of the tangent line at the point (1, 1) is:

And since we know that it passes through the point (1, 1), by the point-slope form:

If desired, we can simplify this into slope-intercept form. Therefore, our equation is:

Problem 18)

We have the equation:

And we want to find the equation of the tangent line to the graph at the point (1, π/4).

Take the derivative of both sides with respect to x:

We can use the chain rule:

Let u(x) = tan⁻¹(x) and let v(x) = x³. Thus:

(Recall that d/dx [arctan(x)] = 1 / (1 + x²).)

Substitute and simplify. Hence:

Then the slope of the tangent line at the point (1, π/4) is:

Then by the point-slope form:

Or in slope-intercept form:

your answer would be 12