find an equation of the tangent line to the curve at the given point:
y=sqrt(1+x^3), (2,3)
To find the equation of the tangent line at a given point on a curve, we need to find the slope of the tangent line at that point and then use the point-slope form of a line to write its equation. Here are the steps to find the equation of the tangent line:
Step 1: Differentiate the function:
Differentiate the given function, y = √(1 + x^3), with respect to x to find its derivative/differential coefficient.
dy/dx = (d/dx) √(1 + x^3)
Step 2: Evaluate the derivative at the given point:
Substitute the x-coordinate of the given point (2,3) into the derivative obtained above to find the slope of the tangent line at that point.
Find dy/dx at x = 2: dy/dx = (d/dx) √(1 + x^3) = ...
Step 3: Use the point-slope form of a line to write the equation:
The equation of a line in point-slope form is given by: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
Substitute the slope found in Step 2, along with the coordinates of the given point (2,3), into the point-slope form to obtain the equation of the tangent line.
y - 3 = m(x - 2)
Finally, simplify the equation if desired to get the final form of the tangent line equation.