Yes, the derivative of an equation is the slope of a line tangent to the graph.
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Take the derivative of the function.
the derivative of tangent dy/dx [ tan(u) ]= [sec^(2)u]u' this means that the derivative of tangent of u is secant squared u times the derivative of u.
Yes. Slope = rise/run (or in calculus, the derivative, often denoted dy/dx). A slope of zero indicates there is no rise, i.e., a horizontal line (or in the case of a curve, the tangent line at a given point is horizontal).
The slope of a curved line changes as you go along the curve and so you may have a different slope at each point. Any any particular point, the slope of the curve is the slope of the straight line which is tangent to the curve at that point. If you know differential calculus, the slope of a curved line at a point is the value of the first derivative of the equation of the curve at that point. (Actually, even if you don't know differential calculus, the slope is still the value of the function's first derivative at that point.)
The gradient of the tangents to the curve.