# Successive Differentiation

### Successive Differentiation:

• Successive differentiation involves finding derivatives of a function $y=f\left(x\right)$ with respect to $x$ multiple times, generating higher-order derivatives.

### Notation:

• First derivative: $\frac{dy}{dx}={f}^{\mathrm{\prime }}\left(x\right)$
• Second derivative: $\frac{{d}^{2}y}{d{x}^{2}}={f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)$
• $n$th derivative: $\frac{{d}^{n}y}{d{x}^{n}}$

### Steps for Successive Differentiation:

1. Find the first derivative of the function $y=f\left(x\right)$.
2. If necessary, find higher-order derivatives by differentiating the function again and again, $n$ times.
3. 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 $n!=n\cdot \left(n-1\right)\cdot \left(n-2\right)\cdot \dots \cdot 2\cdot 1$.
• 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: $\left(uv{\right)}^{\mathrm{\prime }}={u}^{\mathrm{\prime }}v+u{v}^{\mathrm{\prime }}$
• Quotient Rule: ${\left(\frac{u}{v}\right)}^{\mathrm{\prime }}=\frac{{u}^{\mathrm{\prime }}v-u{v}^{\mathrm{\prime }}}{{v}^{2}}$
• Chain Rule: Derivative of a composite function: $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$

### Successive Differentiation Process:

1. Find the first derivative of the function using differentiation rules.
2. Repeat the differentiation process to find higher-order derivatives, applying the derivative rules iteratively.

### Example:

Given $y={x}^{3}+2{x}^{2}+4x+6$:

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

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

3. Find the third derivative: $\frac{{d}^{3}y}{d{x}^{3}}=\frac{d}{dx}\left(6x+4\right)=6$

### Example:

Given $y=3{x}^{4}+2{x}^{3}-5{x}^{2}+7x-9$, find ${y}^{\mathrm{\prime }\mathrm{\prime }}$ and ${y}^{\left(3\right)}$.

1. Find the first derivative:

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

• ${y}^{\mathrm{\prime }\mathrm{\prime }}=\frac{d}{dx}\left(12{x}^{3}+6{x}^{2}-10x+7\right)$
• ${y}^{\mathrm{\prime }\mathrm{\prime }}=36{x}^{2}+12x-10$
3. Find the third derivative ${y}^{\left(3\right)}$:

• ${y}^{\left(3\right)}=\frac{d}{dx}\left(36{x}^{2}+12x-10\right)$
• ${y}^{\left(3\right)}=72x+12$

Interpretation:

• The first derivative (${f}^{\mathrm{\prime }}\left(x\right)$ gives the slope of the function.
• The second derivative (${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)$ represents the rate of change of the slope.
• The third derivative (${f}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\left(x\right)$ 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:
• Higher-order 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 higher-order derivatives.

### Special Considerations:

• Interpretation of Higher Derivatives:

• Beyond the second derivative, higher-order derivatives have specific interpretations in terms of the function's behavior.
• Notation and Symbols:

• The notation for higher-order derivatives involves adding primes ('), exponents, or using $\frac{{d}^{n}}{d{x}^{n}}$ notation.