Chapter 4: Q7P (page 201)

Given c = sin(a - b ), $b = a e^{2 a}$ , find $\frac{d c}{d b}$ .

Short Answer

Expert verified

The $\frac{d c}{d b}$ is $\cos (a - b) (1 - 2 b - e^{2 a})$ .

Step by step solution

Explanation of solution

The provided expressions are $c = \sin (a - b), b = a e^{2 a}$ .

Chain Rule

The number of functions in the composition affects how many differentiation steps are required when using the chain rule to get the derivative of composite functions.

Calculation

Find $\frac{d b}{d a}$ .

$\frac{d b}{d a} = a e^{2 a} (2) + e^{2 a} = 2 a e^{2 a} + e^{2 a}$

Find $\frac{d c}{d a}$ .

$\frac{d c}{d a} = \cos (a - b) (1 - \frac{d b}{d a}) = \cos (a - b) (1 - 2 a e^{2 a} - e^{2 a}) \frac{d c}{d a} = \cos (a - b) (1 - 2 b - e^{2 a})$

Hence the $\frac{d c}{d a}$ is $\cos (a - b) (1 - 2 b - e^{2 a})$ .

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Given c = sin(a - b ), $b = a e^{2 a}$ , find $\frac{d c}{d b}$ .

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

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Given c = sin(a - b ),b=ae2a, find dcdb.

Short Answer

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Given c = sin(a - b ), $b = a e^{2 a}$ , find $\frac{d c}{d b}$ .