AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Calculate dy/dx for sin(x+y) = 3x with Implicit Differentiation
Mathematics
Grade 11 (Senior High School)
Question Content
Find $\frac{dy}{dx}$ for $\\sin(x+y) = 3x$
Correct Answer
$\\frac{dy}{dx} = \\frac{3}{\\cos(x+y)} - 1$
Detailed Solution Steps
1
Step 1: Differentiate both sides with respect to $x$. Use the chain rule on the left side: $\\cos(x+y)\\left(1 + \\frac{dy}{dx}\\right)$, and differentiate the right side to get $3$. This gives $\\cos(x+y)\\left(1 + \\frac{dy}{dx}\\right) = 3$.
2
Step 2: Divide both sides by $\\cos(x+y)$ to isolate the parenthetical term: $1 + \\frac{dy}{dx} = \\frac{3}{\\cos(x+y)}$.
3
Step 3: Subtract 1 from both sides to solve for $\\frac{dy}{dx}$: $\\frac{dy}{dx} = \\frac{3}{\\cos(x+y)} - 1$.
Knowledge Points Involved
1
Implicit Differentiation
Used to differentiate equations where $y$ is not isolated, treating $y$ as a function of $x$ and applying chain rule to terms with $y$.
2
Chain Rule for Trigonometric Functions
When differentiating a composite trigonometric function like $\\sin(x+y)$, the derivative is the derivative of the outer function ($\\cos(x+y)$) multiplied by the derivative of the inner function $(1 + \\frac{dy}{dx})$.
3
Derivative of Sine Function
The derivative of $\\sin(u)$ with respect to $x$ is $\\cos(u) \\cdot \\frac{du}{dx}$, where $u$ is a function of $x$. Here, $u = x+y$.
Loading solution...
