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Question

# Consider the function f defined by $f(x)=5 x-\tan x, 0 \leq x<\frac{\pi}{2}$.a. Find the equation of the tangent to the graph of $f$ at $x=\frac{\pi}{4}$.b. There is a point A on the graph of f where the normal to the graph is vertical. Determine the coordinates of A.

Solution

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(a) Differentiate $f(x)$ with respect to $x$ to evaluate $f'(x)$.

\begin{aligned}f'(x)&=\dfrac{d}{dx}[f(x)]\\&=\dfrac{d}{dx}(5x-\tan{x})\\&=\dfrac{d}{dx}(5x)-\dfrac{d}{dx}(\tan{x})&&\color{#4257b2}\text{Difference rule}\\&=5\dfrac{d}{dx}(x)-\dfrac{d}{dx}(\tan{x})&&\color{#4257b2}\text{Constant Multiple rule}\\&=5x^{1-1}-\sec^2{x}\\&=5-\sec^2{x}\end{aligned}

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