Find an equation of the tangent line to the curve $y = sin(3x) + sin^2 (3x)$ given the point (0,0). Answer is $y = 3x$, but please explain solution steps.
[Math] Find an equation of the tangent line to the curve at the given point. y = sin(3x) sin2 (3x) given the point (0,0)
calculusderivativestangent line
Related Question
- [Math] Find equation of a line perpendicular to the tangent of curve at a given point.
- [Math] how to find tangent line at a given point, without equation
- [Math] For a curve, find the unit tangent vector and parametric equation of the line tangent to the curve at the given point
- [Math] Find tangent equation for a curve which is perpendicular to a line
Best Answer
Hint:
Do you know that the slope of the tangent line at a point of the graph of a function is the derivative of the function at this point?
So, for $y = \sin(3x) + \sin^2 (3x)$ find the derivative $y'$ ( can you do?), then evaluate this derivative for $x=0$
Now the line has equation $y=mx$ with $m=y'(0)$
Using the chain rule the derivative is: $$ y'=\cos(3x)\cdot(3x)'+2\sin(3x)(\sin(3x))'=3\cos(3x)+2\sin(3x)\cos(3x)(3x)'$$$$=3\cos(3x)+6\sin(3x)\cos(3x) $$ so $y'(0)=3$.