Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![# Multiple Derivatives
A function \( f \) is \( n \)-times differentiable (or \( f \in C^n \)) if you can apply the derivative \( n \) times to \( f \) and have a continuous function after each application of the derivative.
A function \( f \) is smooth (or \( f \in C^\infty \)) if it can be differentiated infinitely many times, and each derivative is a continuous function.
## Derivative Properties
**Linearity:**
\[
\frac{d}{dx} [f(x) + a \cdot g(x)] = f'(x) + a \cdot g'(x)
\]
**Products:**
\[
\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]
**Quotients:**
\[
\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}, \quad \text{where } g(x) \ne 0
\]
**Compositions:**
\[
\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
\]
## Tangent and Normal Lines
If \( y = f(x) \) describes some differentiable function, the equation of the tangent line at a point is given by
\[
y = f'(a)(x - a) + f(a).
\]
The equation of the normal line at a point is given by
\[
y = -\frac{1}{f'(a)}(x - a) + f(a).
\]
## Linear Approximation
If \( f \) is differentiable near \( a \), then for values \( x \) close to \( a \),
\[
f(x) \approx f(a)(x - a) + f(a).
\]
# Basic Function Derivatives
- **Constant:**
\[
\frac{d}{dx} [a] = 0, \quad \text{where } a
\]
- **Power:**
\[
\frac{d}{dx} [x^r] = r \cdot](https://content.bartleby.com/qna-images/question/f4b0f97a-89e5-4466-a6e2-73ccf61a13a9/3ab0f606-1d2c-4166-be4e-b6cdf34992e9/pmb4kt7_processed.png)
Transcribed Image Text:# Multiple Derivatives
A function \( f \) is \( n \)-times differentiable (or \( f \in C^n \)) if you can apply the derivative \( n \) times to \( f \) and have a continuous function after each application of the derivative.
A function \( f \) is smooth (or \( f \in C^\infty \)) if it can be differentiated infinitely many times, and each derivative is a continuous function.
## Derivative Properties
**Linearity:**
\[
\frac{d}{dx} [f(x) + a \cdot g(x)] = f'(x) + a \cdot g'(x)
\]
**Products:**
\[
\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]
**Quotients:**
\[
\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}, \quad \text{where } g(x) \ne 0
\]
**Compositions:**
\[
\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
\]
## Tangent and Normal Lines
If \( y = f(x) \) describes some differentiable function, the equation of the tangent line at a point is given by
\[
y = f'(a)(x - a) + f(a).
\]
The equation of the normal line at a point is given by
\[
y = -\frac{1}{f'(a)}(x - a) + f(a).
\]
## Linear Approximation
If \( f \) is differentiable near \( a \), then for values \( x \) close to \( a \),
\[
f(x) \approx f(a)(x - a) + f(a).
\]
# Basic Function Derivatives
- **Constant:**
\[
\frac{d}{dx} [a] = 0, \quad \text{where } a
\]
- **Power:**
\[
\frac{d}{dx} [x^r] = r \cdot
Transcribed Image Text:### Calculus Problems
**5. Determine the equations of the lines tangent to the curve \( x^2y + 2xy^2 = x + 6y \) when \( x = 2 \).**
For this problem, you'll need to find the derivative of the implicit function to determine the slope of the tangent line at the specified point \( x = 2 \). Then, use the point-slope form of the equation of a line to find the tangent line equations.
---
**6. Use linear approximation to estimate the value of \( \log(99) \). To how many decimal places is the linear approximation accurate?**
In this task, apply the concept of linear approximation around a nearby easy-to-calculate value, such as \( \log(100) \), and find the linear approximation of \( \log(99) \). Determine the accuracy by comparing it to the actual value.
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