2.3: Differentiation Rules
- Page ID
- 204095
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2.3 Differentiation Rules
- Apply the constant, power, sum and difference, product, and quotient rules to calculate the derivatives of a variety of functions, including polynomials, rational functions, and products of functions.
- Use the extended power rule to find the derivatives of functions with negative exponents, demonstrating the correct application to both positive and negative integer exponents.
- Combine the constant, power, sum and difference, product, and quotient rules to calculate the derivatives of polynomial and rational functions.
Differentiation Formulas:
1. Derivative of a constant function:
\[
\frac{d}{dx}(c) = 0
\]
2. Power rule (for any real number \( n \)):
\[
\frac{d}{dx}(x^n) = nx^{n - 1}
\]
3. Constant multiple rule:
\[
\frac{d}{dx}[c f(x)] = c \frac{d}{dx} f(x)
\]
4. Sum rule:
\[
\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)
\]
5. Difference rule:
\[
\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx} f(x) - \frac{d}{dx} g(x)
\]
6. Product rule:
\[
\frac{d}{dx}[f(x)g(x)] = \frac{d}{dx}[f(x)] \cdot g(x) + f(x) \cdot \frac{d}{dx}[g(x)]
\]
7. Quotient rule:
\[
\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right]
= \frac{ \frac{d}{dx}[f(x)] \cdot g(x) - f(x) \cdot \frac{d}{dx}[g(x)] }{ [g(x)]^2 }
\]
Find the derivative of each function below.
- \( y = 7 \)
- \( y = x^5 \)
- \( y = 4x^9 - x^2 + 1 \)
- \( y = x^\pi \)
- \( y = x^2 - \dfrac{3}{x} \)
- \( y = x^5(3x^2 + 2x + 1) \)
- \( y = x^4 \left( \dfrac{10}{x^4} + \dfrac{1}{x^2} \right) \)
- \( y = \dfrac{8x^6}{3x^2 - 1} \)
- \( y = \dfrac{\sqrt{x} + 1}{x^3 - 4x^2 + 5} \)
Find the equation of the tangent line \( T(x) \) to the graph of the given function at the specified point.
(1) \( f(x) = x^2 - \dfrac{3}{x} \) at \( (3, 8) \)
(2) \( f(x) = x^4 \left( \dfrac{10}{x^4} + \dfrac{1}{x^2} \right) \) at \( (2, 14) \)
(3) \( f(x) = \dfrac{\sqrt{x} + 1}{x^3 - 4x^2 + 5} \) at \( (1, 1) \)
Determine all points on the graph of
\[
f(x) = x^3 - 6x^2 +9x
\]
for which:
(a) the tangent line is horizontal
(b) the tangent line has a slope of \( -3 \)