Help Derivatives plz bouta drop this class (Answers and explain plz you trying to tell me what I should know versus with what I do is not helpful at all)

Question

Find the equation of the tangent line to the curve y = x⁴+6e^x at the point (0,6)

y = _________

Answer 1

One of the main uses of the derivative is to FIND THE SLOPE of the TANGENT LINE.

To find the slope of the tangent line, find the derivative of the function, then plug in the x-value they gave you.

For this problem, y = x⁴ + 6e^x the derivative would be (dy/dx) = 4x^3 + 6e^x.

They asked for the tangent line at the point (0,6), so the x-coordinate is 0. Plug in 0 for all of the x's in the derivative. The resulting number will be the slope of the tangent line. slope = y '(0) = (dy/dx) at x = 0 is

y'(0) = 4(0)³ + 6 e⁰ = 0 + 6 = 6 So the slope is 6.

So now we now the slope of the line whose equation we are trying to write. We also know one point. And this is all that is needed to write the equation of a line.

Point-Slope Form: y-y1 = m(x-x1) for us would be y-6 = 6(x - 0) or y - 6 = 6x and this simplifies to the form requested by adding the 6 to both sides.

y = 6x + 6 is the equation of the tangent line at (0,6)