Podcast
Questions and Answers
Suppose $f(x)$ is differentiable at $x = a$. Which of the following statements is MOST accurate concerning the relationship between differentiability and continuity of $f(x)$ at $x = a$?
Suppose $f(x)$ is differentiable at $x = a$. Which of the following statements is MOST accurate concerning the relationship between differentiability and continuity of $f(x)$ at $x = a$?
- If $f(x)$ is continuous at $x = a$, then $f(x)$ is differentiable at $x = a$.
- If $f(x)$ is not continuous at $x = a$, then $f(x)$ is differentiable at $x = a$.
- If $f(x)$ is differentiable at $x = a$, then $f(x)$ is not necessarily continuous at $x = a$.
- If $f(x)$ is differentiable at $x = a$, then $f(x)$ is continuous at $x = a$. (correct)
The derivative of a function at a point represents the slope of the secant line to the function at that point.
The derivative of a function at a point represents the slope of the secant line to the function at that point.
False (B)
Determine the derivative of $f(x) = c$, where $c$ is a constant. Write your answer as a simple equation.
Determine the derivative of $f(x) = c$, where $c$ is a constant. Write your answer as a simple equation.
$f'(x) = 0$
The derivative of a linear function $f(x) = ax + b$ is equal to the __________ of x.
The derivative of a linear function $f(x) = ax + b$ is equal to the __________ of x.
Given $f(x) = x^n$, where $n$ is a real number, determine $f'(x)$:
Given $f(x) = x^n$, where $n$ is a real number, determine $f'(x)$:
Given $h(t) = 8t^3 - \frac{2}{5t^2} - t + 12$, what is $h'(t)$?
Given $h(t) = 8t^3 - \frac{2}{5t^2} - t + 12$, what is $h'(t)$?
Given $I(s) = \sqrt{s} + 9\sqrt[3]{s^2} - \frac{4}{\sqrt{s^3}}$, what is $I'(s)$?
Given $I(s) = \sqrt{s} + 9\sqrt[3]{s^2} - \frac{4}{\sqrt{s^3}}$, what is $I'(s)$?
Given $f(x) = \sqrt{x}(5x^4 + 10x - 72)$, compute $f'(x)$. Pick the closest match:
Given $f(x) = \sqrt{x}(5x^4 + 10x - 72)$, compute $f'(x)$. Pick the closest match:
Calculate the derivative of $g(x) = \frac{8x^2 - 5x}{x^2} - 2x^{-2}$:
Calculate the derivative of $g(x) = \frac{8x^2 - 5x}{x^2} - 2x^{-2}$:
Given $y = (4x^2 - 5)(8x - 7)$, find the derived function:
Given $y = (4x^2 - 5)(8x - 7)$, find the derived function:
Determine $y'$ for $y = \sqrt[4]{x^3}(4x^2 - 5)$:
Determine $y'$ for $y = \sqrt[4]{x^3}(4x^2 - 5)$:
Differentiate $y = (\frac{4}{x} + 1)(x^2 - 2x + 3)$ to find $y'$:
Differentiate $y = (\frac{4}{x} + 1)(x^2 - 2x + 3)$ to find $y'$:
Find the derivative of $y = (1 + \sqrt{x})(\sqrt[5]{x^2} - \sqrt[3]{x})$:
Find the derivative of $y = (1 + \sqrt{x})(\sqrt[5]{x^2} - \sqrt[3]{x})$:
Given $y = \frac{4x^2 - 5}{8x - 7}$, find $y'$:
Given $y = \frac{4x^2 - 5}{8x - 7}$, find $y'$:
Compute the derivative of $y = \frac{8\sqrt{x}}{1 - 2x^4}$:
Compute the derivative of $y = \frac{8\sqrt{x}}{1 - 2x^4}$:
Determine the derived function of $y = \frac{1 + 2x^4}{1 - 2x^4}$:
Determine the derived function of $y = \frac{1 + 2x^4}{1 - 2x^4}$:
Given the function $f(x) = (x^3 - 3)^{100}$, find $f'(x)$.
Given the function $f(x) = (x^3 - 3)^{100}$, find $f'(x)$.
Determine the derivative of $f(x) = \sqrt{x^4 - 2x^3 + 2x - 11}$.
Determine the derivative of $f(x) = \sqrt{x^4 - 2x^3 + 2x - 11}$.
Let $y = (x^2 - 1)^4(x^3 + 1)^8$. Find $y'$ in simplified form.
Let $y = (x^2 - 1)^4(x^3 + 1)^8$. Find $y'$ in simplified form.
Determine the derivative $\frac{dy}{dx}$ of $y = \frac{(x^2 + 2x + 3)^3}{(2x^3 - 1)^2}$.
Determine the derivative $\frac{dy}{dx}$ of $y = \frac{(x^2 + 2x + 3)^3}{(2x^3 - 1)^2}$.
If the volume of water in a tank is given by $V(t) = 3t^2 - 12t + 6$, then at $t=2$ minutes the volume of water will be constant.
If the volume of water in a tank is given by $V(t) = 3t^2 - 12t + 6$, then at $t=2$ minutes the volume of water will be constant.
To differentiate a sum or difference of functions, one must differentiate individual terms and then put them back together with opposite signs.
To differentiate a sum or difference of functions, one must differentiate individual terms and then put them back together with opposite signs.
Using the limit definition of the derivative, determine the derivative of $f(x) = 5x^2 + 3x - 2$.
Using the limit definition of the derivative, determine the derivative of $f(x) = 5x^2 + 3x - 2$.
Express the chain rule using Leibniz notation for $y = f(u)$ and $u = g(x)$.
Express the chain rule using Leibniz notation for $y = f(u)$ and $u = g(x)$.
Match each function type with its corresponding derivative rule.
Match each function type with its corresponding derivative rule.
Assuming the function $f(x)$ is not differentiable at $x=c$, what must be true concerning $f(x)$ at $x=c$?
Assuming the function $f(x)$ is not differentiable at $x=c$, what must be true concerning $f(x)$ at $x=c$?
The derivative of $f(x) + g(x)$ is equal to $f'(x) + g'(x) + f(x)g(x)$
The derivative of $f(x) + g(x)$ is equal to $f'(x) + g'(x) + f(x)g(x)$
The ______ rule enables the calculation of derivatives for composite functions.
The ______ rule enables the calculation of derivatives for composite functions.
What does the formula $f'(x) = lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ represent?
What does the formula $f'(x) = lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ represent?
Given the function $f(x) = 20x^5 - 25x^4 + 10x - 71$, what is its derived function $f'(x)$?
Given the function $f(x) = 20x^5 - 25x^4 + 10x - 71$, what is its derived function $f'(x)$?
According to the sum and difference rules, the derivative $(f(x) + g(x))'$ always equals $f'(x) - g'(x)$
According to the sum and difference rules, the derivative $(f(x) + g(x))'$ always equals $f'(x) - g'(x)$
Given the functions, match the application of derivative rule.
Given the functions, match the application of derivative rule.
Suppose $f(x) = \frac{a}{x} + bx^2$, where $a$ and $b$ are arbitrary real numbers. What are the conditions under which $f'(1) = 0$?
Suppose $f(x) = \frac{a}{x} + bx^2$, where $a$ and $b$ are arbitrary real numbers. What are the conditions under which $f'(1) = 0$?
If $f(x)=x^{2}+2x+5$ apply the definition of the derivative to determine $f'(x)$. Show your algebraic work.
If $f(x)=x^{2}+2x+5$ apply the definition of the derivative to determine $f'(x)$. Show your algebraic work.
If $y = x^{x}$ then $\frac{dy}{dx} = x^{x}$
If $y = x^{x}$ then $\frac{dy}{dx} = x^{x}$
Consider the function $f(x) = |x|$. Which statement is most accurate?
Consider the function $f(x) = |x|$. Which statement is most accurate?
Which rule is applied while computing the derived function $y = \frac{u}{v}$, where $u$ equals $f(x)$ and $v$ equals $g(x)$?
Which rule is applied while computing the derived function $y = \frac{u}{v}$, where $u$ equals $f(x)$ and $v$ equals $g(x)$?
According to the properties of differentiation, the derivative of $cf(x)$ equals $c$ multiplied by the derivative of ____.
According to the properties of differentiation, the derivative of $cf(x)$ equals $c$ multiplied by the derivative of ____.
Flashcards
Differentiation
Differentiation
The process of finding a derivative.
Derivative
Derivative
A limit used to find the slope of the tangent line to a function.
Differentiable Function
Differentiable Function
f(x) is differentiable at x=a if f'(a) exists.
Derivative of a Constant
Derivative of a Constant
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Derivative of a Linear Function
Derivative of a Linear Function
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The Sum and Difference Rule
The Sum and Difference Rule
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Constant Multiple Rule
Constant Multiple Rule
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Power Rule
Power Rule
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Product Rule
Product Rule
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Quotient Rule
Quotient Rule
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Chain Rule
Chain Rule
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Study Notes
Derivatives
- Derivatives find the slope of a line tangent to a function at a point.
- Derivatives have uses across different disciplines like velocity and acceleration in physics marginal profit roles in business, and growth rates in biology.
- Finding the derivative is called differentiation.
- The derivative of f(x) with respect to x is f'(x) when the limit exists.
Derivative Definition
- f ‘(x) = lim (f(x + h) - f(x)) / h as h approaches 0.
- A function, f(x), is differentiable at x = a if f '(a) exists.
- f(x) is differentiable on an interval if the derivative exists for each point in that internal.
- If f(x) is differentiable at x=a then f(x) is continuous at x=a.
Derivatives as Rate of Change
- A derivative signifies a rate of change.
- If f(x) is a quantity then f '(a) is the instantaneous rate of change of f(x) at x = a.
- Amount of water in a tank at t minutes is given by V(t) = 3t² - 12t + 6.
- Rate of change at t=1 is V'(1) = 6(1) – 12 = − 6.
- At t = 1 the rate of change is negative which means the in tank is decreasing.
- Rate of change at t=5 is V'(5) = 6(5) – 12 = 18.
- At t=5 the rate of change is positive which means the volume in tank is increasing.
- Volume is not changing at V’(t)=0 or 6t – 12 = 0, and t will equal 2 minutes.
Constant Derivation
- The function f(x) always equals a constant c, thus f(x) = c and f(x + h) = c.
- f'(x) = lim (f(x+h)-f(x)) / h = lim (c-c) / h = 0/h == 0 as h approaches 0.
Linear Function Derivation
- The function f(x) equals ax + b, thus f(x) always equals ax + b and f(x + h) = a(x + h) + b = ax + ah + b
- f'(x) = lim (f(x+h)-f(x)) / h = lim (ax+ah+b-(ax+b)) / h = (ah) / h= a as h approaches 0.
- The derivative of a linear fucntion is always a constant and equals the coefficient of x.
Quadratic Function Derivation
- Let the function f(x) equal ax², where a is a constant.
- f(x) = ax2 and f(x + h) = a(x + h)2 = a(x² + 2xh + h²) = ax2 + 2axh + ah²
- f'(x) = lim (f(x+h)-f(x)) / h = lim (ax² + 2axh + ah²-ax2) / h = (2axh + ah²) / h =2ax + ah as h approaches 0
- f′(x) = 2ax since h approaches 0
Sum and Difference Rules
- The sum and difference rules allow derivatives of multiple term functions.
- (f(x) ± g(x))' = f '(x) ± g'(x).
Constant Multiple Rule
- (cf(x))' = cf '(x).
- The derivative of a constant times a function is the constant times the derivative.
Constant Derivative
- If f(x) = c, then f'(x) = 0.
- The derivative of a constant is zero.
Power Rule
- If f(x) = xn, then f'(x) = nxn-1, 1, n represents any number.
Product Rule
- If f(x) and g(x) are differentiable, then the product is differentiable.
- (f (x)g(x))' = f (x)g'(x) + f '(x)g (x)
Quotient Rule
- If f(x) and g(x) are differentiable, then the quotient is differentiable.
- (f(x) / g(x))’ = (g(x)f'(x) − f(x)g'(x)) / [g(x)]2
Chain Rule
- If f(x) and g(x) are differentiable, and F(x) = (f°g)(x), then the derivative of F(x) is: F'(x) = f '(g(x))g'(x)
- if y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx)
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