Rules of Differentiation: Calculus Study Notes

Rules of Differentiation

Constant Rule

The Constant Rule states that the derivative of any constant function is zero. This is because a constant function does not change, so its rate of change is zero.

• Formula: $(c)' = 0$

• Example: $(28)' = 0$

Power Rule

The Power Rule is used to differentiate functions of the form $x^n$, where $n$ is a real number. The rule states that the derivative is $n$ times $x$ raised to the power $n-1$.

• Formula: $(x^n)' = n x^{n-1}$

• Example: $(x^9)' = 9x^{8}$

Constant Multiple Rule

The Constant Multiple Rule states that the derivative of a constant times a function is the constant times the derivative of the function.

• Formula: $(c \cdot f(x))' = c \cdot f'(x)$

• Example: $(c \cdot x^n)' = c \cdot n x^{n-1}$

• Example: $(c \cdot x)' = c$

Sum and Difference Rule

The Sum/Difference Rule states that the derivative of a sum or difference of functions is the sum or difference of their derivatives.

• Formula: $(f(x) \pm g(x))' = f'(x) \pm g'(x)$

Derivative of the Exponential Function

The derivative of the exponential function $e^x$ is itself.

• Formula: $(e^x)' = e^x$

Examples of Differentiation

Applying the above rules to various functions:

• Example 1: $f(x) = -5x^{20}$ $f'(x) = -5 \cdot 20 x^{19} = -100x^{19}$

• Example 2: $g(x) = 7x$ $g'(x) = 7$

• Example 3: $f(x) = 6\sqrt{x} = 6x^{1/2}$ $f'(x) = 6 \cdot \frac{1}{2} x^{-1/2} = 3x^{-1/2}$

• Example 4: $f(x) = 12x^5 - 40x + e^x - e^4$ $f'(x) = 60x^4 - 40 + e^x$

• Example 5: $y = 5x^{-3} + 4x^{-1}$ $y' = -15x^{-4} - 4x^{-2}$

Application: Slope of Tangent Line

To find the point on a curve where the slope of the tangent line equals a given value, set the derivative equal to that value and solve for $x$.

• Example: For $f(x) = 6x^2 - 8x$, find where the slope is 16. $f'(x) = 12x - 8$ Set $12x - 8 = 16$ $x = 2$ $f(2) = 8$ Point: (2, 8)

Application: Horizontal Tangent Lines

Horizontal tangent lines occur where the derivative is zero.

• Example: For $f(x) = x^3 + 4x^2 - 3x$ $f'(x) = 3x^2 + 8x - 3$ Set $3x^2 + 8x - 3 = 0$ Solutions: $x = -3, \frac{1}{3}$

Application: Equation of Tangent Line

To find the equation of the tangent line at a point, use the point-slope form with the derivative evaluated at the given $x$ value.

• Example: For $f(x) = x^3 + 4x^2 - 3x$ at (1, 2): $f'(x) = 3x^2 + 8x - 3$ $f'(1) = 8$ Equation: $y - 2 = 8(x - 1) \Rightarrow y = 8x - 6$

Higher Order Derivatives

The second derivative measures the rate of change of the first derivative, often related to concavity and acceleration. The nth derivative is the result of differentiating a function $n$ times.

• Second Derivative Notation: $f''(x) = (f'(x))'$ or $\frac{d^2f}{dx^2}$

• nth Derivative Notation: $f^{(n)}(x) = (f^{(n-1)}(x))'$ or $\frac{d^n f}{dx^n}$

Example: First and Second Derivatives

Find the first and second derivatives of a polynomial function.

• Example: $f(x) = 2x^3 + 9x^2 + 6x$ $f'(x) = 6x^2 + 18x + 6$ $f''(x) = 12x + 18$