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Questions and Answers
What is the derivative of a function represented by?
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?
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?
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?
How is the slope of a line derived from two points on it?
What does the limit as Δx approaches zero represent?
What does the limit as Δx approaches zero represent?
Which two mathematicians are mentioned in relation to derivatives?
Which two mathematicians are mentioned in relation to derivatives?
What is represented by the symbols Δy and Δx in calculus?
What is represented by the symbols Δy and Δx in calculus?
What does a derivative describe when considering a curve?
What does a derivative describe when considering a curve?
What is the method used to find the derivative of a function?
What is the method used to find the derivative of a function?
What is the derivative of a constant function?
What is the derivative of a constant function?
If a function is defined as g(x) = x^2, what is the derivative of this function?
If a function is defined as g(x) = x^2, what is the derivative of this function?
Which rule states that the derivative of a product of two functions is the product of the derivatives plus the product of the functions?
Which rule states that the derivative of a product of two functions is the product of the derivatives plus the product of the functions?
What is the derivative of a power function represented as $f'(x) = x^b$?
What is the derivative of a power function represented as $f'(x) = x^b$?
What does the quotient rule for differentiation help to find?
What does the quotient rule for differentiation help to find?
If $y = (x^2 + c) + (ax^4 + b)$, what is the derivative $y'$?
If $y = (x^2 + c) + (ax^4 + b)$, what is the derivative $y'$?
What is the correct interpretation of the limit in the context of finding the derivative?
What is the correct interpretation of the limit in the context of finding the derivative?
What is the derivative of the function $y = v^2 - 3v + 2$ with respect to $x$ if $v = 4x^2 + 1$?
What is the derivative of the function $y = v^2 - 3v + 2$ with respect to $x$ if $v = 4x^2 + 1$?
Using the power rule, how would you differentiate $y = (x + a)^3$?
Using the power rule, how would you differentiate $y = (x + a)^3$?
What expression represents $dy/dx$ for parametric equations $x = 2t + 3$ and $y = t^2 - 1$?
What expression represents $dy/dx$ for parametric equations $x = 2t + 3$ and $y = t^2 - 1$?
What is $d^2y/dx^2$ given $dy/dx = 3x(x + a)^2$?
What is $d^2y/dx^2$ given $dy/dx = 3x(x + a)^2$?
For the function $y = x^3 + a$, what is the first derivative with respect to $x$?
For the function $y = x^3 + a$, what is the first derivative with respect to $x$?
What is the second derivative of the function $y = v^2$ if $v = x^2 + 1$?
What is the second derivative of the function $y = v^2$ if $v = x^2 + 1$?
How do you differentiate the function $y = (x + a)^2 + 5bx - cx$?
How do you differentiate the function $y = (x + a)^2 + 5bx - cx$?
If $v = 3x^2 + a^2$, what is the first derivative $dv/dx$?
If $v = 3x^2 + a^2$, what is the first derivative $dv/dx$?
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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
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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.
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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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