Calculate the derivative of xy + sin(x) = e^y with respect to x

Published on October 20, 2024

Calculus

The derivative of xy + sin(x) = e^y with respect to x is found using implicit differentiation, applying the product rule and chain rule, and solving for dy/dx.

Question

Answer

Let’s break down how to find the derivative of

with respect to

1. Implicit Differentiation

Since we have

mixed in with

terms, we’ll use implicit differentiation. This means we’ll differentiate both sides of the equation with respect to

, treating

as a function of

2. Applying the Derivative Rules

Product Rule: For the term , we’ll use the product rule:
d(uv)/dx = u(dv/dx) + v(du/dx)
Chain Rule: Since is a function of , we’ll use the chain rule when differentiating terms involving . For example, the derivative of with respect to is .

Step-by-Step Solution

Differentiate both sides:
Apply the product rule to xy:
Isolate dy/dx terms:
Factor out dy/dx:
Solve for dy/dx:

Therefore, the derivative of xy + sin(x) = e^y with respect to x is:

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