Maths Calculus Differentiation Problem

(1). Steps:

a) Find the slope of the straight line. x - y + 8 = 0. Let's call this m.

b) Differentiate y = x/1 + x. dy/dx will give you the slope of tangent to the curve for any value of x.

c) Solve for dy/dx = m, i.e. for what values of x, do you get dy/dx = m? This is your x coordinates of the points where the tangent is parallel to the line x - y + 8 = 0.

d) Substitute these values of x into the initial equation y = x/1 + x to get the y coordinate(s).

See how far you can get with these steps.

Could also be [latex]\frac{x}{1+x}[/latex]. As the problem is is stated it's y = x / 1 + x which is y = 2x.

OP, please clarify your question with parentheses.

The equation [latex]y=\frac{x}{1 + x}[/latex] is almost certainly what the OP meant as it has two solutions for x where the differentiated function equals the slope of the straight line. The other interpretation [latex]y=\frac{1}{x}+x[/latex] doesn't have any solutions.

Here's the updated set of steps for the OP to follow for the first part of the question.

(1). Steps:

a) Find the slope of the straight line. [latex]x - y + 8 = 0[/latex] by expressing it in the form [latex]y=mx+c[/latex] where [latex]m[/latex] is the slope.

b) Differentiate [latex]y=\frac{x}{1+x}[/latex] with respect to [latex]x[/latex]. [latex]\frac{dy}{dx}[/latex] will give you the slope of tangent to the curve for any value of x.

c) Solve for [latex]\frac{dy}{dx}=m[/latex], i.e. for what values of [latex]x[/latex], do you get dy/dx = m? This is your [latex]x[/latex]-coordinates of the points where the tangent is parallel to the line [latex]x - y + 8 = 0[/latex].

d) Substitute these values of [latex]x[/latex] into the initial equation [latex]y=\frac{x}{1+x}[/latex] to get the [latex]y[/latex]-coordinates for the two values of [latex]x[/latex].

See how far you can get with these steps.