The Quotient Rule: Not Just Dividing Derivatives But Simple Enough
The quotient rule for derivatives allows calculation of the derivative of a function divided by another. It is essential because the derivative of a quotient of two functions isn’t simply the quotient of their derivatives, necessitating a distinct formula for accurate differentiation in various applications.
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Definition:
To use quotient rule, you subtract the product of the bottom function and the derivative of the top from the product of the top and the derivative of the bottom, then divide it all by the bottom function squared. Here is the mathematical formula for the quotient rule:
\( \left(\frac{f}{g}\right)’ = \frac{f’g – fg’}{g^2} \)
Example 1:
Let’s solve an example.
\( f(x) = \sin x, \ g(x) = x^2 + 1\)
\( f'(x) = \cos x, \ g'(x) = 2x \)
\(\Rightarrow \left(\frac{\sin x}{x^2 + 1}\right)’ = \frac{\cos x \cdot (x^2 + 1) – \sin x \cdot 2x}{(x^2 + 1)^2} \)
\( = \frac{\cos x \cdot x^2 + \cos x – 2x \sin x}{(x^2 + 1)^2} \)
Example 2:
\( f(x) = x^3, \ g(x) = \cos x \)
\(f'(x) = 3x^2, \ g'(x) = -\sin x \)
\(\Rightarrow \left(\frac{x^3}{\cos x}\right)’ = \frac{3x^2 \cdot \cos x – x^3 \cdot (-\sin x)}{\cos^2 x} \)
\( = \frac{3x^2 \cos x + x^3 \sin x}{\cos^2 x} \)
Hints:
- In some complex fractions, applying logarithmic differentiation simplifies the process more than the quotient rule would.
- For \( \frac{1}{x} \) and \( \frac{1}{f(x)} \), we use the following formulas, although \( \frac{1}{x} \) could be solved using power rule too.
\( \left(\frac{1}{x}\right)’ = -\frac{1}{x^2} \)
\( \left(\frac{1}{f(x)}\right)’ = -\frac{f'(x)}{[f(x)]^2} \)
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