Differentiate a Polynomial Using the Chain Rule

Differentiate $(2x^2 + 4x - 3)^{199}$ using the Chain Rule.

In this problem, you are tasked with differentiating a function of the form $(2x^2 + 4x - 3)^{199}$ using the Chain Rule. The Chain Rule is a fundamental technique in calculus used to differentiate compositions of functions. When a function is composed of an outer function and an inner function, the chain rule provides a systematic way of taking the derivative of such compositions by multiplying the derivative of the outer function by the derivative of the inner function.

This specific problem is an excellent exercise in applying the chain rule because it involves both polynomial differentiation and the handling of large exponents. The high power of the expression underscores the importance of recognizing the outer function, $u(x)^{199}$ in this case, where $u(x)$ is the inner function $2x^2 + 4x - 3$. By differentiating the outer function and utilizing the power rule within the context of the chain rule, you can simplify the task of differentiating complex composite functions quickly.

Mastering problems like this enhances your understanding of differentiation techniques and prepares you for tackling more intricate calculus problems, including those involving exponential, trigonometric, and logarithmic functions. This conceptual clarity is fundamental for progressing through more advanced calculus topics.