Find each indicated derivative and simplify. h^'(t) if h(t)=(-0.05 t^2)/(2 t+1)

Name: Problem 84, Chapter 3: Calculus for Business, Economics, Life Sciences, and Social Sciences
Uploaded: 2021-02-19T20:52:22-08:00
Duration: 1 min 54 s
Channel: Gregory Higby
Description: Find each indicated derivative and simplify. h^'(t) if h(t)=(-0.05 t^2)/(2 t+1)

step 1 Okay, we're looking at the quotient negative 0 .05 t squared over 2T plus 1, and we're asked to

Find each indicated derivative and simplify. h^'(t) if ...

Key Concepts

Differentiation

Differentiation is the process of finding the rate at which a function changes at any point, often represented as the derivative. It forms a central concept in calculus and is applied to functions to determine slopes of tangent lines, rates of change, or instantaneous velocity.

Power Rule

The power rule is a basic technique in differentiation that states if a function has the form f(x) = x^n, then its derivative is f'(x) = n*x^(n–1). This rule is essential for differentiating polynomial functions and is applied to individual terms of a polynomial.

Constant Multiple Rule

The constant multiple rule in differentiation indicates that a constant factor multiplying a function can be pulled out of the differentiation process. In other words, if f(x) is differentiable and c is a constant, then the derivative of c*f(x) is c multiplied by the derivative of f(x).

Quotient Rule

The quotient rule is used to differentiate functions that are expressed as the ratio of two differentiable functions. It states that if a function is given by h(x) = f(x)/g(x), then the derivative h'(x) is given by (f'(x)g(x) – f(x)g'(x)) divided by [g(x)]^2. This rule is particularly useful for handling rational functions.

Simplification

Simplification involves algebraic manipulation to put the derivative in a more reduced and interpretable form. After differentiating, expressions are often simplified to cancel common factors or to combine like terms, making the final format more concise and easier to use for further analysis.

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