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Chapter 3 | Derivatives
Finding the Equation of a Tangent Line
Find the equation of a line tangent to the graph of f ( x ) =cot x at x = π 4 .
Solution To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute f ⎛ ⎝ π 4 ⎞ ⎠ =cot π 4 =1. Thus the tangent line passes through the point ⎛ ⎝ π 4 , 1 ⎞ ⎠ . Next, find the slope by finding the derivative of f ( x ) =cot x and evaluating it at π 4 : f ′( x ) =−csc 2 x and f ′ ⎛ ⎝ π 4 ⎞ ⎠ =−csc 2 ⎛ ⎝ π 4 ⎞ ⎠ =−2. Using the point-slope equation of the line, we obtain y −1=−2 ⎛ ⎝ x − π 4 ⎞ ⎠
or equivalently,
y =−2 x +1+ π 2 .
Example 3.44 Finding the Derivative of Trigonometric Functions
Find the derivative of f ( x ) =csc x + x tan x .
Solution To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find f ′( x ) = d dx (csc x )+ d dx ( x tan x ). In the first term, d dx (csc x ) =−csc x cot x , and by applying the product rule to the second term we obtain d dx ( x tan x ) = (1)(tan x )+(sec 2 x )( x ). Therefore, we have f ′( x ) =−csc x cot x +tan x + x sec 2 x .
Find the derivative of f ( x ) =2tan x −3cot x .
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