Differentiate the function \( g(x)=2 x^{2}+3 \tan x \)

To differentiate the function \( g(x) = 2x^2 + 3\tan x \), you take the derivative of each term separately. The derivative of \( 2x^2 \) is \( 4x \), and the derivative of \( 3\tan x \) is \( 3\sec^2 x \). So, putting it all together, the derivative is: \[ g'(x) = 4x + 3\sec^2 x. \] Now you have a formula to find the slope of the tangent line to the curve at any point \( x \)! Understanding how this applies in real-world contexts can be a game changer! For instance, in physics, this differentiation can represent speed when dealing with position functions that track the motion of objects. If you know the position of an object as a function of time \( x \), differentiating it provides you with its velocity, allowing you to predict movements or implement strategies for avoiding obstacles. When differentiating functions, one common mistake is forgetting to apply the chain rule when dealing with composite functions. For example, if the argument of \( \tan \) were more complex, say \( \tan(3x) \), you would need to multiply by the derivative of the inner function, which is \( 3 \). Always double-check your chain rules to avoid losing those crucial terms!