One of the applications that I can think of has to do with physics. In fact, almost everything you learn in differential calculus is applied to physics. For example, the motion of an object attached to a spring can be modeled using a sine wave, as follows: 

To find the velocity of this mass, we need to take the derivative of a y = sin x graph. The derivative would be y = cos x. Then, to find the acceleration of the mass, we need to take the derivative of y = cos x, which would be y = -sin x. So, finding the derivative of a cosine function can be applied to understanding the motion of a mass on a spring.