27–76. Calculate the derivative of the following functions.
$y={(3x^2+7x)}^{10}$

Verified step by step guidance

1. Step 1: Recognize that the function y = (3x^2 + 7x)^{10} is a composite function, which means we will use the chain rule to find its derivative.
2. Step 2: Identify the outer function and the inner function. Here, the outer function is u^{10} and the inner function is u = 3x^2 + 7x.
3. Step 3: Differentiate the outer function with respect to the inner function u. The derivative of u^{10} with respect to u is 10u^9.
4. Step 4: Differentiate the inner function u = 3x^2 + 7x with respect to x. The derivative is 6x + 7.
5. Step 5: Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function: dy/dx = 10(3x^2 + 7x)^9 * (6x + 7).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that represents the slope of the tangent line to the curve of the function at any given point. Derivatives are used to find rates of change, optimize functions, and analyze the behavior of functions.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when differentiating functions raised to a power, as seen in the given problem.

Power Rule

The power rule is a basic rule for finding the derivative of a function in the form of x raised to a power. It states that if f(x) = x^n, then the derivative f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for polynomial functions and is essential for calculating derivatives of terms within more complex functions.