Select Page

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)} \$