Successive Differentiation
Successive Differentiation:
 Successive differentiation involves finding derivatives of a function $y=f(x)$ with respect to $x$ multiple times, generating higherorder derivatives.
Notation:
 First derivative: $\frac{dy}{dx}={f}^{\mathrm{\prime}}(x)$
 Second derivative: $\frac{{d}^{2}y}{d{x}^{2}}={f}^{\mathrm{\prime}\mathrm{\prime}}(x)$
 $n$th derivative: $\frac{{d}^{n}y}{d{x}^{n}}$
Steps for Successive Differentiation:
 Find the first derivative of the function $y=f(x)$.
 If necessary, find higherorder derivatives by differentiating the function again and again, $n$ times.
 Interpret the derivatives in terms of the function's behavior at different levels of rate of change.
Rules for Successive Differentiation:
 Power Rule: The $n$th derivative of ${x}^{n}$ is $n!=n\cdot (n1)\cdot (n2)\cdot \dots \cdot 2\cdot 1$.
 Constant Rule: The derivative of a constant $c$ is always zero.
 Sum and Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives.
 Product Rule: $(uv{)}^{\mathrm{\prime}}={u}^{\mathrm{\prime}}v+u{v}^{\mathrm{\prime}}$
 Quotient Rule: ${\left(\frac{u}{v}\right)}^{\mathrm{\prime}}=\frac{{u}^{\mathrm{\prime}}vu{v}^{\mathrm{\prime}}}{{v}^{2}}$
 Chain Rule: Derivative of a composite function: $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$
Successive Differentiation Process:
 Find the first derivative of the function using differentiation rules.
 Repeat the differentiation process to find higherorder derivatives, applying the derivative rules iteratively.
Example:
Given $y={x}^{3}+2{x}^{2}+4x+6$:

Find the first derivative: $\frac{dy}{dx}=3{x}^{2}+4x+4$

Find the second derivative: $\frac{{d}^{2}y}{d{x}^{2}}=\frac{d}{dx}(3{x}^{2}+4x+4)=6x+4$

Find the third derivative: $\frac{{d}^{3}y}{d{x}^{3}}=\frac{d}{dx}(6x+4)=6$
Example:
Given $y=3{x}^{4}+2{x}^{3}5{x}^{2}+7x9$, find ${y}^{\mathrm{\prime}\mathrm{\prime}}$ and ${y}^{(3)}$.

Find the first derivative:
 ${y}^{\mathrm{\prime}}=12{x}^{3}+6{x}^{2}10x+7$

Find the second derivative ${y}^{\mathrm{\prime}\mathrm{\prime}}$:
 ${y}^{\mathrm{\prime}\mathrm{\prime}}=\frac{d}{dx}(12{x}^{3}+6{x}^{2}10x+7)$
 ${y}^{\mathrm{\prime}\mathrm{\prime}}=36{x}^{2}+12x10$

Find the third derivative ${y}^{(3)}$:
 ${y}^{(3)}=\frac{d}{dx}(36{x}^{2}+12x10)$
 ${y}^{(3)}=72x+12$
Interpretation:
 The first derivative (${f}^{\mathrm{\prime}}(x)$ gives the slope of the function.
 The second derivative (${f}^{\mathrm{\prime}\mathrm{\prime}}(x)$ represents the rate of change of the slope.
 The third derivative (${f}^{\mathrm{\prime}\mathrm{\prime}\mathrm{\prime}}(x)$ describes the rate of change of the rate of change of the slope or the concavity of the curve.
Applications of Successive Differentiation:
 Rate of Change Analysis:
 Higherorder derivatives provide information on changes in rate of change, acceleration, and curvature of functions.
 Optimization Problems:
 Identifying critical points (where derivatives are zero) to determine maximum or minimum values.
 Physics and Engineering:
 Studying motion, acceleration, and forces by examining higherorder derivatives.
Special Considerations:

Interpretation of Higher Derivatives:
 Beyond the second derivative, higherorder derivatives have specific interpretations in terms of the function's behavior.

Notation and Symbols:
 The notation for higherorder derivatives involves adding primes ('), exponents, or using $\frac{{d}^{n}}{d{x}^{n}}$ notation.
Advantages of Successive Differentiation:
 Understanding Complex Functions:
 Higherorder derivatives reveal more detailed information about a function's behavior.
 Problem Solving:
 Useful in solving differential equations and modeling dynamic systems.