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AP Calculus AB/BC26Updated Sep 29, 20263 pages

Mastering Differentiation Rules for Calculus

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Seleena Busi@seleena_busi07
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Differentiation is a powerful tool in calculus that helps us find rates of change. These rules provide techniques for differentiating various functions, allowing you to tackle everything from simple polynomials to complex quotients without starting from first principles each time.
Rules of Differentiation – page 1

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Rules for Differentiation

Ever wondered how to calculate rates of change quickly? The rules of differentiation make this possible. The derivative of any constant is always 0 - this makes sense as constants don't change as x changes!

When dealing with powers of x, the power rule tells us that ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}. This lets you find derivatives of polynomials easily. For example, if p = t³ + 6t² - 5/t + 16, then the derivative is dp/dt = 3t² + 12t + 5/t².

The product rule handles functions multiplied together. If you have two functions u and v, then ddx(uv)=uv′+vu′\frac{d}{dx}(uv) = uv' + vu'. For example, when differentiating x² + 1$$x + 3, you first identify u = x² + 1 and v = x + 3, find their derivatives, and apply the formula.

Quick Tip: Horizontal tangent lines occur exactly where the derivative equals zero. This is extremely useful when sketching graphs or finding maximum/minimum points!

Rules of Differentiation – page 2

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The Quotient Rule

When one function is divided by another, you need the quotient rule. If you have uv\frac{u}{v}, the derivative is ddx(uv)=u′v−vu′v2\frac{d}{dx}(\frac{u}{v}) = \frac{u'v - vu'}{v^2}. It might look complicated, but it's just a formula to memorize.

Let's see it in action: for f(x)=x2−1x2+1f(x) = \frac{x^2 - 1}{x^2 + 1}, we identify u = x² - 1 and v = x² + 1. After finding their derivatives u′=2x,v′=2xu' = 2x, v' = 2x and applying the quotient rule, we get f′(x)=4x(x2+1)2f'(x) = \frac{4x}{(x^2 + 1)^2}.

You can also use these rules with numerical values. For instance, if you know specific values of functions and their derivatives at a point, you can calculate the derivative of their product or quotient at that point without finding the general formula first.

Remember: The quotient rule has a specific pattern: "low d-high minus high d-low, over the square of what's below." This memory aid might help you recall the formula during tests!

Rules of Differentiation – page 3

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Higher Order Derivatives

Higher order derivatives help us analyze how rates of change themselves are changing. Finding them is straightforward - just differentiate repeatedly!

Take y = x³ - 5x² + 2. The first derivative y' = 3x² - 10x shows the rate of change. The second derivative y'' = 6x - 10 reveals how that rate is itself changing. Continuing, y''' = 6 (a constant) and y⁴ = 0.

Notice how each derivative becomes simpler until we reach zero. This happens with all polynomials - the nth derivative of an n-degree polynomial is always a constant, and all higher derivatives are zero.

Interesting fact: Higher order derivatives have practical applications in physics - acceleration is the second derivative of position, and jerk (the rate of change of acceleration) is the third derivative!

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