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.



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 $y'(0)=3$.

  • $\begingroup$ I know that I have to find the derivative, but I don't know how to. I also know that once you find m (by taking the derivative) you can plug it into the $y-y=m(x-x)$ or $y = mx + b$ slope formulas. $\endgroup$ – Chaniqua Ranson Oct 5 '16 at 2:04
  • $\begingroup$ Added to my answer. $\endgroup$ – Emilio Novati Oct 5 '16 at 7:40


Find the slope $m$, and intercept $b$, for the line $$ y = mx + b, $$ tangent at the origin to the curve $$ f(x) = \sin ^2(3 x)+\sin (3 x). $$

Intercept $b$

Because the function goes through the origin, the tangent line will also go through the origin. Therefore the $y-$intercept $b=0$.

Slope $m$

The slope of the tangent line $m$ is, by definition, the same as the slope of the target function at the point of contact. The slope of the function is $$ f'(x) = 6 \sin (3 x) \cos (3 x) + 3 \cos (3 x). $$ The problem specifies $x_{*} = 0.$ The point of contact is $$ \left( x_{*}, f(x_{*}) \right) = \left( 0, 3 \right). $$ The slope of the function at the point of contact is $$ f'( x_{*} ) = 6 \sin (3 x_{*}) \cos (3 x_{*}) + 3 \cos (3 x_{*}) = 6\cdot 0 \cdot 0 + 3 \cdot 1 = 3. $$ The slope of the tangent line is $$ m = f'( x_{*} ) = 3. $$

Solution The equation of the line tangent to $f(x)$ at $x_{*} = 0$ is $$ y = mx + b = 3x. $$

Tangent line


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.