Here's another one - I think that I am starting to understand it.
Find dy/dx by implicit differentiation:
x^7 = cot y
My answer is csc^2y / 7x^6
7x^6 dx=-csc y dy
Did you miss the sign?
I did miss the sign but would the answer be: 7x^6
- -----
csc^2y
correct.
Great job on continuing to practice and improve your understanding! Implicit differentiation can be a bit trickier than explicit differentiation, but with practice, it becomes easier.
To find dy/dx using implicit differentiation, we follow these steps:
1. Start by differentiating both sides of the equation with respect to x.
- For the left side, we apply the power rule: d/dx(x^n) = nx^(n-1).
- For the right side, we'll use the chain rule since y is a function of x.
2. Differentiate the left side:
- The derivative of x^7 with respect to x is 7x^(7-1), which simplifies to 7x^6.
3. Differentiate the right side:
- To differentiate cot y, we use the chain rule. The derivative of cot u with respect to x is -csc^2u du/dx.
- In our case, u is y. du/dx is dy/dx, which is what we're trying to find.
Therefore, the derivative of cot y is -csc^2y dy/dx.
4. Set the two derivatives equal to each other:
7x^6 = -csc^2y dy/dx
5. Solve for dy/dx:
- To isolate dy/dx, divide both sides of the equation by -csc^2y.
- The equation becomes dy/dx = (7x^6) / -csc^2y
Now, we can simplify further. Remember that csc^2y is the reciprocal of sin^2y. So, we can rewrite it as 1/sin^2y.
Therefore, dy/dx = 7x^6 / - (1/sin^2y)
To simplify further, we multiply the numerator and denominator of the fraction by sin^2y:
dy/dx = -7x^6sin^2y
So, the final answer is dy/dx = -7x^6sin^2y.
Keep up the great work! Practice like this will help solidify your understanding of implicit differentiation.