Derivatives Find and simplify the derivative of the following functions.
g(t) = 3t² + 6/t⁷

Verified step by step guidance

1. Step 1: Identify the function components. The function is g(t) = 3t^2 + 6/t^7. It consists of two terms: 3t^2 and 6/t^7.
2. Step 2: Rewrite the function for easier differentiation. The term 6/t^7 can be rewritten using negative exponents as 6t^(-7). So, g(t) = 3t^2 + 6t^(-7).
3. Step 3: Differentiate each term separately. Use the power rule for differentiation, which states that the derivative of t^n is n*t^(n-1).
4. Step 4: Apply the power rule to the first term. The derivative of 3t^2 is 2*3*t^(2-1) = 6t.
5. Step 5: Apply the power rule to the second term. The derivative of 6t^(-7) is -7*6*t^(-7-1) = -42t^(-8).

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative is often denoted as f'(x) or df/dx, and it can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Power Rule

The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions, making it easier to compute derivatives quickly.

Quotient Rule

The Quotient Rule is a method for differentiating functions that are expressed as the ratio of two other functions. If a function is defined as f(x) = u(x)/v(x), where both u and v are differentiable, the derivative is given by (v*u' - u*v')/v². This rule is essential when dealing with rational functions, allowing for the correct computation of their derivatives.