**Step 1:** In the preferred kind of the item regulation formula, we will certainly note the very first multiplicand as $latex u$ and also the 2nd multiplicand as $latex v$.

For that reason, we have

$ latex u = wrong {(x)} $

$ latex v = tan {(x)} $

**Step 2:** Discover the by-products of $latex u$ and also $latex v$:

$ latex u’ = cos {(x)} $

$ latex v’ = sec ^ {2} {(x)} $

**Step 3:** Use the formula for the item regulation replacing $latex u$, $latex u’$, $latex v$ and also $latex v’$.

$ latex frac {d} {dx} (uv) = uv’ + vu’$

$$ frac {d} {dx} (uv) = (wrong {(x)}) cdot (sec ^ {2} {(x)} )+ (tan {(x)}) cdot (cos {(x)} )$$

**Step 4:** Simplify algebraically and also considering that we have a trigonometric feature in our by-product, we can additionally use some relevant trigonometric identifications in our option:

$$ frac {d} {dx} (uv) = wrong {(x)} sec ^ {2} {(x)} + tan {(x)} cos {(x)} $$

$$ frac {d} {dx} (uv) = (wrong {(x)}) (frac {1} {cos {(x)}} )^ 2+ (frac {wrong {(x)}} {cos {(x)}}) (cos {(x)} )$$

$$ frac {d} {dx} (uv) = (wrong {(x)}) (frac {1 ^ {2}} {cos ^ {2} {(x)}} )+ (frac {wrong {(x)}} {cos {(x)}}) (cos {(x)} )$$

$$ frac {d} {dx} (uv) = (frac {wrong {(x)}} {cos {(x)}}) (frac {1} {cos {(x)}} )+ (frac {wrong {(x)}} {cos {(x)}}) (cos {(x)} )$$

$$ frac {d} {dx} (uv) = sec {(x)} tan {(x)} + wrong {(x)} $$

$ latex f'( x) = sec {(x)} tan {(x)} + wrong {(x)} $