Introduction to Function Derivatives

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Questions and Answers

What is the derivative of a function represented by?

f'(x) or df/dx (correct)

dy only

f(x) only

d/dx of f(x)

What does the derivative of a function indicate?

The function's minimum value

The rate of change of the function (correct)

The area under the curve

The maximum value of the function

What is the correct interpretation of dy/dx?

A single term indicating a variable relationship (correct)

An equation of a line

An unrelated mathematical term

A fraction representing a slope

How is the slope of a line derived from two points on it?

By finding the change in x and y Signup and view all the answers

What does the limit as Δx approaches zero represent?

The slope of the tangent to the curve Signup and view all the answers

Which two mathematicians are mentioned in relation to derivatives?

Newton and Leibniz Signup and view all the answers

What is represented by the symbols Δy and Δx in calculus?

The changes in the y and x coordinates Signup and view all the answers

What does a derivative describe when considering a curve?

The slope of the tangent line at a specific point Signup and view all the answers

What is the method used to find the derivative of a function?

Differentiation Signup and view all the answers

What is the derivative of a constant function?

0 Signup and view all the answers

If a function is defined as g(x) = x^2, what is the derivative of this function?

2x Signup and view all the answers

Which rule states that the derivative of a product of two functions is the product of the derivatives plus the product of the functions?

Product Rule Signup and view all the answers

What is the derivative of a power function represented as $f'(x) = x^b$?

$b x^{b-1}$ Signup and view all the answers

What does the quotient rule for differentiation help to find?

The derivative of ratios Signup and view all the answers

If $y = (x^2 + c) + (ax^4 + b)$, what is the derivative $y'$?

$2x + 4ax^3$ Signup and view all the answers

What is the correct interpretation of the limit in the context of finding the derivative?

The instantaneous rate of change Signup and view all the answers

What is the derivative of the function $y = v^2 - 3v + 2$ with respect to $x$ if $v = 4x^2 + 1$?

$8x(2v - 3)$ Signup and view all the answers

Using the power rule, how would you differentiate $y = (x + a)^3$?

$3(x + a)^2$ Signup and view all the answers

What expression represents $dy/dx$ for parametric equations $x = 2t + 3$ and $y = t^2 - 1$?

$rac{dy}{dt} / rac{dx}{dt}$ Signup and view all the answers

What is $d^2y/dx^2$ given $dy/dx = 3x(x + a)^2$?

$6x(x + a)$ Signup and view all the answers

For the function $y = x^3 + a$, what is the first derivative with respect to $x$?

$3x^2$ Signup and view all the answers

What is the second derivative of the function $y = v^2$ if $v = x^2 + 1$?

$8x$ Signup and view all the answers

How do you differentiate the function $y = (x + a)^2 + 5bx - cx$?

$2(x + a) + 5b - c$ Signup and view all the answers

If $v = 3x^2 + a^2$, what is the first derivative $dv/dx$?

$6x$ Signup and view all the answers

Study Notes

Introduction to Function Derivatives

The derivative of a function f(x) represents its rate of change.
It is denoted by either f'(x) or df/dx.
The derivative of a function can be denoted by both f'(x) and dy/dx.
dy/dx is a single term, not a fraction.
It is read as the derivative of a function y with respect to x.

Slope of a Curve

The slope of a line is calculated by dividing the change in y (Δy) by the change in x (Δx).
The slope of a line is constant at every point.
The slope of a curve at a point is represented by the slope of the tangent line at that point.
To find the slope of the tangent at a point P on a curve, we move a second point Q along the curve towards P.
As Q approaches P, the slope of the line connecting P and Q approaches the slope of the tangent at P.
In the limit as Δx approaches zero, the slope of the tangent to the curve at the point (x, y) is given by the derivative of y with respect to x (dy/dx).

Differentiation

Differentiation is the method used to find the derivative of a function.
Differentiation examples include finding the derivative of m(x) = 2x + 5, g(x) = x², and h(x) = x³.

Principal Rules of Differentiation

Derivative of a Constant Function:
- If y = c (where c is a constant), then f'(x) = 0.
Derivative of a Constant Times a Function:
- d(c * v)/dx = c * dv/dx.
Derivative of a Power Function:
- If f(x) = xᵇ, then f'(x) = b * x^(b-1).
Derivative of a Polynomial Function:
- d(u ± v)/dx = du/dx ± dv/dx.
Derivative of a Product:
- d(u * v)/dx = u * dv/dx + v * du/dx.
Derivative of a Quotient:
- d(u/v)/dx = (v * du/dx - u * dv/dx)/v².
Differentiation of a Function of a Function (Chain Rule):
- dy/dx = dy/dv * dv/dx.

Differentiation of Parametric Equations

For parametric equations x = f(t) and y = g(t), the derivative dy/dx can be found by using the chain rule:
- dy/dx = (dy/dt) / (dx/dt).

Examples

Example 1: Find Dx y if y = x² at any point.
- The value of the function at any point P is y = x².
- The value of the function at a point P' is y = (x+Δx)².
- Δy = (x + Δx)² - x² = 2xΔx + (Δx)².
- Dividing by Δx, we get Δy/Δx = 2x + Δx.
- Taking the limit as Δx approaches zero, we get Dx y = lim (2x + Δx) = 2x.
Example 2: Use the power rule to find Dxy where y = (x² + a)²³.
- Let v = x² + a.
- y = (v)³.
- dy/dv = 3v².
- dv/dx = 2x.
- Using the chain rule, dy/dx = dy/dv * dv/dx = 3v² * 2x = 3(x² + a)² * 2x = 6x(x² + a)².

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Description

This quiz covers the fundamentals of function derivatives, including the definition and representation of derivatives, the concept of slope, and the process of differentiation. Test your understanding of how derivatives represent the rate of change of functions and how to calculate the slope of curves.