Product & Quotient Rules # 7

Question

Find the equation of the line that is tangent to the curve

at the point .

The equation of this tangent line can be written in the form  where

 

and  

Answer 1

To find the slope, take the derivative to find y' then plug in x = π to find the slope at (π, -5π)

To find the derivative, since y is (5x) multiplied by cos(x), use the product rule:

the derivative of u•v = u'v + uv'

u = 5x

v = cos(x)

u' = 5

v' = -sin(x)

So y' = (5)(cos(x)) + (5x)(-sin(x)) = 5cos(x) - 5xsin(x)

y'(π) = 5cos(π) - 5πsin(π) = 5(-1) - 5π(0) = -5

Using the point-slope form of a line [y - y₁ = m(x - x₁)] the equation of the line is:

y - (-5π) = -5(x - π)

y + 5π = -5x + 5π

y = -5x

so m = -5 and b = 0