Covers the basic notation for derivatives (Lagrange, Leibniz), the Constant Rule, the Constant Multiple Rule, the Sum/Difference Rule, and the fundamental Power Rule for finding derivatives of functions like x^n. This stack builds the essential groundwork for all shortcut differentiation.
What are the two primary notations for the derivative of a function \(y = f(x)\)?
Lagrange's notation: \(f'(x)\) (read as "f prime of x").
Leibniz's notation: \(\frac{dy}{dx}\) (read as "dee y, dee x").
State the Constant Rule for differentiation. What is the derivative of \(f(x) = c\), where \(c\) is a constant?
The derivative of any constant function is \(0\).
$$ \frac{d}{dx}(c) = 0 $$
This is because the graph of \(f(x)=c\) is a horizontal line with a slope of \(0\).
State the Constant Multiple Rule. How do you find the derivative of \(c \cdot f(x)\), where \(c\) is a constant?
The derivative of a constant times a function is the constant times the derivative of the function.
$$ \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) $$
Find the derivative of \(f(x) = -10\).
$$ f'(x) = 0 $$
This is an application of the Constant Rule, as \(-10\) is a constant.
State the Power Rule for differentiation for a function of the form \(f(x) = x^n\).
To find the derivative, bring the exponent \(n\) down as a coefficient and subtract \(1\) from the original exponent.
$$ \frac{d}{dx}(x^n) = n x^{n-1} $$
Use the Power Rule to find the derivative of \(f(x) = x^7\).
$$ f'(x) = 7x^{7-1} = 7x^6 $$
Use the Power Rule to find the derivative of \(f(x) = x\).
First, rewrite \(f(x) = x\) as \(f(x) = x^1\). Then apply the Power Rule:
$$ f'(x) = 1 \cdot x^{1-1} = 1 \cdot x^0 = 1 \cdot 1 = 1 $$
Find the derivative of \(g(x) = 5x^4\).
Using the Constant Multiple Rule and Power Rule:
$$ g'(x) = 5 \cdot \frac{d}{dx}(x^4) = 5 \cdot (4x^3) = 20x^3 $$
State the Sum and Difference Rules for derivatives. What is \(\frac{d}{dx}[f(x) \pm g(x)]\)?
The derivative of a sum or difference of functions is the sum or difference of their individual derivatives.
$$ \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) $$
Find the derivative of the polynomial \(f(x) = 3x^2 + 5x - 9\).
Apply the rules to each term separately:
$$ f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) - \frac{d}{dx}(9) = 6x + 5 - 0 = 6x + 5 $$
How do you apply the Power Rule to a radical function like \(f(x) = \sqrt{x}\)?
First, rewrite the radical using a fractional exponent: \(f(x) = x^{1/2}\). Then you can apply the Power Rule.
Find the derivative of \(f(x) = \sqrt{x}\).
Rewrite as \(f(x) = x^{1/2}\). Then, using the Power Rule:
$$ f'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} $$
Find the derivative of \(g(t) = 4\sqrt[3]{t}\).
Rewrite as \(g(t) = 4t^{1/3}\). Then differentiate:
$$ g'(t) = 4 \cdot \frac{1}{3}t^{\frac{1}{3}-1} = \frac{4}{3}t^{-2/3} = \frac{4}{3\sqrt[3]{t^2}} $$
How do you apply the Power Rule to a function like \(h(x) = \frac{1}{x^3}\)?
First, rewrite the function using a negative exponent: \(h(x) = x^{-3}\). Then you can apply the Power Rule.
Find the derivative of \(h(x) = \frac{1}{x^3}\).
Rewrite as \(h(x) = x^{-3}\). Then, using the Power Rule:
$$ h'(x) = -3x^{-3-1} = -3x^{-4} = -\frac{3}{x^4} $$
Find \(\frac{dy}{dx}\) for \(y = 6x^3 - \frac{1}{2x^4} + 1\).
First, rewrite the function as \(y = 6x^3 - \frac{1}{2}x^{-4} + 1\). Then differentiate term by term:
$$ \frac{dy}{dx} = 18x^2 - \frac{1}{2}(-4x^{-5}) + 0 = 18x^2 + 2x^{-5} = 18x^2 + \frac{2}{x^5} $$
