Differentiation of xcosx and Probability of Drawing Specific Cards from a Deck
<p>The image displays two separate questions. I will provide the solutions for both.</p>
<p>For the differentiation of \( x\cos{x} \) with respect to \( x \) using the first principle:</p>
<p>We have \( f(x) = x\cos{x} \), we need to find \( f'(x) \) using the first principle:</p>
<p>\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]</p>
<p>\[ = \lim_{{h \to 0}} \frac{(x+h)\cos(x+h) - x\cos{x}}{h} \]</p>
<p>We then expand and arrange the expression and apply the limit. However, without the options or further context for finding the value of \( k \), this part of the question is incomplete.</p>
<p>For the probability question, assuming we are looking to find the probability of drawing 1 diamond and 3 spades:</p>
<p>The total number of ways to draw 4 cards from a 52 card deck is \( C(52, 4) \).</p>
<p>The number of ways to draw 1 diamond from the 13 available diamonds is \( C(13, 1) \).</p>
<p>The number of ways to draw 3 spades from the 13 available spades is \( C(13, 3) \).</p>
<p>The probability \( P \) of the event is:</p>
<p>\[ P = \frac{C(13, 1) \cdot C(13, 3)}{C(52, 4)} \]</p>
<p>\[ P = \frac{13 \cdot \frac{13!}{3!(13-3)!}}{\frac{52!}{4!(52-4)!}} \]</p>
<p>We can then simplify the factorials to get the probability.</p>
