find d2y/dx2 in terms of x and y
Understand the Problem
The question is asking for the second derivative of y with respect to x, denoted as d²y/dx², which involves taking the derivative of the first derivative dy/dx. The task will require understanding how y relates to x, potentially involving implicit differentiation if y is a function of x.
Answer
$$ \frac{d^2y}{dx^2} = 2 $$
Answer for screen readers
The final answer is: $$ \frac{d^2y}{dx^2} = 2 $$
Steps to Solve
- Find the first derivative ($dy/dx$)
Assuming that you have an explicit function for $y$, take the derivative of $y$ with respect to $x$. If $y$ is given implicitly, you would use implicit differentiation.
Example: If $y = x^2 + 3x$, then: $$ \frac{dy}{dx} = 2x + 3 $$
- Differentiate to find the second derivative ($d²y/dx²$)
Now, take the derivative of the first derivative $dy/dx$ with respect to $x$. This will give you the second derivative.
Continuing from the previous example: $$ \frac{d^2y}{dx^2} = \frac{d}{dx}(2x + 3) = 2 $$
- Final Expression for the Second Derivative
Combine the results to express your final answer for the second derivative.
In this case, the second derivative is: $$ \frac{d^2y}{dx^2} = 2 $$
The final answer is: $$ \frac{d^2y}{dx^2} = 2 $$
More Information
The second derivative gives insight into the concavity of the function and the acceleration (if $y$ represents distance and $x$ represents time). Here, a second derivative of 2 indicates that the function is concave up and the rate of change of the slope is constant.
Tips
- Forgetting to apply the power rule appropriately when finding derivatives.
- Confusing the differentiation order; ensure correct symbols and functions are used.
- Not simplifying the expressions fully before deriving.
