Question

# Find the derivatives of the given functions. g(x)=(\sin x)(\tan x)

Derivatives
Find the derivatives of the given functions. $$\displaystyle{g{{\left({x}\right)}}}={\left({\sin{{x}}}\right)}{\left({\tan{{x}}}\right)}$$

2021-06-21
$$\displaystyle{g{{\left({x}\right)}}}={\left({\sin{{x}}}\right)}{\left({\tan{{x}}}\right)}$$
$$\displaystyle{g}'{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\left({\sin{{x}}}\right)}{\left({\tan{{x}}}\right)}\right)}$$
$$\displaystyle={\sin{{x}}}\dot{{\lbrace}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\tan{{x}}}\right)}+{\tan{{x}}}\dot{{\lbrace}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{x}}}\right)}$$
$$\displaystyle={\sin{{x}}}\dot{{\lbrace}}{{\sec}^{{{2}}}{x}}+{\tan{{x}}}\dot{{\lbrace}}{\cos{{x}}}$$
$$\displaystyle={\sin{{x}}}\dot{{\lbrace}}{{\sec}^{{{2}}}{x}}+{\frac{{{\sin{{x}}}}}{{{\cos{{x}}}}}}\dot{{\cos{{x}}}}$$
$$\displaystyle={\sin{{x}}}\dot{{\lbrace}}{{\sec}^{{{2}}}{x}}+{\sin{{x}}}$$
$$\displaystyle={\sin{{x}}}{\left({1}+{{\sec}^{{{2}}}{x}}\right.}$$