Please help !!!!!
Find dy/dx by implicit differentiation.
(sin πx + cos πy)4 = 65
I assume that the 4 is an exponent, and you mean
(sin ðx + cos ðy)^4 = 65
Differentiate both sides of the equation with respect to x.
4 (sin ðx + cos ðy)^3 *d/dx[sin ðx + cos ðy] = 0
[sin(pi*x) + cos(pi*y)]^3 * [(pi*cos(pi*x) - pi*sin(pi*y)*(dy/dx)] = 0
You can divide out the first [ ] term on the left. Only second [] term can be zero.
(pi*cos(pi*x) = pi*sin(pi*y)*dy/dx
dy/dx = (1/pi)cot(pi*y)
(pi*cos(pi*x) = pi*sin(pi*y)*dy/dx
dy/dx = (1/pi)cot(pi*y)
I think the pi should be canceled, and the term cos and sin can not be identified as cot, because it is cos of x, and sin of y.
Is it right?
pi can be cancelled, and yes, your are right about the cot.
Thank you for your alert corrections
So, What do u think how we solve this? :)
To find dy/dx using implicit differentiation, we will differentiate both sides of the equation with respect to x, treating y as an implicit function of x.
Let's go step by step:
Step 1: Differentiate both sides of the equation with respect to x.
On the left-hand side, we have (sin πx + cos πy)^4. To differentiate this expression, we will use the chain rule.
The chain rule states that if we have a composite function of the form (f(g(x)))^n, the derivative is given by:
(d/dx)[f(g(x)))^n] = n * (f(g(x)))^(n-1) * f'(g(x)) * g'(x).
Let's apply the chain rule to our expression:
d/dx[(sin πx + cos πy)^4] = 4 * (sin πx + cos πy)^3 * (d/dx)(sin πx + cos πy).
On the right-hand side, we have a constant value, 65. The derivative of a constant is always zero.
Therefore, we have:
4 * (sin πx + cos πy)^3 * (d/dx)(sin πx + cos πy) = 0
Step 2: Simplify the equation.
Since the derivative of a constant is zero, we can simplify the equation by dividing both sides by 4 * (sin πx + cos πy)^3:
(d/dx)(sin πx + cos πy) = 0 / (4 * (sin πx + cos πy)^3)
Simplifying further, we have:
(d/dx)(sin πx + cos πy) = 0
Step 3: Solve for (d/dx)(sin πx + cos πy).
The derivative of sin πx with respect to x is π * cos πx. The derivative of cos πy with respect to x is 0 since it does not contain an x term.
Therefore, we have:
π * cos πx + 0 = 0
Simplifying:
π * cos πx = 0
Step 4: Solve for dy/dx.
To solve for dy/dx, we need to isolate the dy/dx term. However, in this case, we cannot solve for dy/dx directly since the equation does not explicitly define y in terms of x.
Therefore, the value of dy/dx remains unknown or undefined in this implicit differentiation problem.
I hope this explanation helps you understand how to find dy/dx using implicit differentiation. Let me know if you have any further questions!