Find dy/dx by implicit differentiation for the following equation. e^x^2=7x+5y+3

1 answer:

5 0

Answer:

$\displaystyle \frac{dy}{dx} = \frac{2e^{2x} - 7}{5}$

General Formulas and Concepts:

Pre-Algebra

Equality Properties

Algebra I

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Implicit Differentiation

Basic Power Rule:

eˣ Derivative: $\displaystyle \frac{d}{dx}[e^u] = u'e^u$

Step-by-step explanation:

Step 1: Define

$\displaystyle (e^x)^2 = 7x + 5y + 3$

Step 2: Differentiate

Rewrite [Exponential Rule - Powering]: $\displaystyle e^{2x} = 7x + 5y + 3$
Implicit Differentiation: $\displaystyle \frac{d}{dx}[e^{2x}] = \frac{d}{dx}[7x + 5y + 3]$
[Derivative] eˣ derivative: $\displaystyle 2e^{2x} = \frac{d}{dx}[7x + 5y + 3]$
[Derivative] Basic Power Rule: $\displaystyle 2e^{2x} = 1 \cdot 7x^{1 - 1} + 1 \cdot 5y^{1 - 1}\frac{dy}{dx}$
[Derivative] Simplify: $\displaystyle 2e^{2x} = 7 + 5\frac{dy}{dx}$
[Derivative] [Subtraction Property of Equality] Subtract 7 on both sides: $\displaystyle 2e^{2x} - 7 = 5\frac{dy}{dx}$
[Derivative] [Division Property of Equality] Divide 5 on both sides: $\displaystyle \frac{2e^{2x} - 7}{5} = \frac{dy}{dx}$
[Derivative] Rewrite: $\displaystyle \frac{dy}{dx} = \frac{2e^{2x} - 7}{5}$