`f(x) = sin(x)(sin(x) + cos(x)), (pi/4,1)` Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

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Chapter 2, 2.3 - Problem 62 - Calculus of a Single Variable (10th Edition, Ron Larson).
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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) = sin x*(sin x + cos x). Use the product rule to determine the derivative of f(x).

f'(x) = (sin x)'*(sin x + cos x) + sin x*(sin x + cos x)'

f'(x) = (cos x)*(sin x + cos x) + sin x*(cos x - sin x)

= `cos^2x + cos x*sin x + sin x*cos x - sin^2x`

= `cos^2 x - sin ^2x + 2*cos x*sin x`

At `x = pi/4` , the value of f'(x) = 1. The tangent at the point `(pi/4, 1)` has slope 1 and the equation of the tangent is : `(y - 1)/(x - pi/4) = 1`

`y = (x - pi/4 + 1)`

This can be seen in the following graph:

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