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:

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

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.

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 $y=\mathrm{sin}(3{x}^{2}+5)$:

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

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

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