# Differentiation by Substitution

### Differentiation by Substitution:

• Differentiation by substitution is a technique used to simplify the process of finding derivatives by replacing complex expressions with simpler variables.
• It's based on the chain rule and often involves substituting one variable or expression with another to make the derivative calculation more manageable.

### Steps for Differentiation by Substitution:

1. Identify a complex part of the function that can be substituted with a simpler variable.

2. Choose an appropriate substitution:

• Select a new variable or expression to replace the complex part of the function.
• Ensure the substitution makes the differentiation process easier.
3. Compute the derivative using the substituted variable:

• Differentiate the function with respect to the new variable.
• Substitute back the original expression to express the derivative in terms of the original variable.

### Example:

Given the function $y=\mathrm{sin}\left(3{x}^{2}+5\right)$:

1. Identify the complex part: $3{x}^{2}+5$

2. Choose a substitution: Let $u=3{x}^{2}+5$.

3. Compute the derivative using substitution:

• Express $y$ as $y=\mathrm{sin}\left(u\right)$.
• Find $\frac{dy}{du}$. $\frac{dy}{du}=\mathrm{cos}\left(u\right)$
• Now, find $\frac{du}{dx}$ : $\frac{du}{dx}=6x$
• Apply the chain rule: $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$. $\frac{dy}{dx}=\mathrm{cos}\left(u\right)\cdot 6x$
• Substitute back $u=3{x}^{2}+5$ and $\mathrm{cos}\left(u\right)=\mathrm{cos}\left(3{x}^{2}+5\right)$ : $\frac{dy}{dx}=6x\cdot \mathrm{cos}\left(3{x}^{2}+5\right)$

### Tips for Substitution in Differentiation:

• Choose effective substitutions:

• Opt for substitutions that simplify the expression and make differentiation easier.
• Chain Rule Application:

• Remember to apply the chain rule when differentiating the substituted expression.
• Substitution Reversal:

• Substitute back the original expression to express the final derivative in terms of the original variable.

### Advantages of Substitution in Differentiation:

• Simplification of Complex Functions:

• It helps handle intricate functions by replacing parts with simpler expressions.
• Ease in Calculation:

• Substitution often makes differentiation more straightforward, especially for functions involving nested or composite structures.

### Special Cases:

• Nested Functions:

• Particularly useful for functions within functions, making nested derivatives more manageable.
• Inverse Functions:

• Substitution can aid in finding derivatives of inverse functions by choosing appropriate substitutions.