if sinh(y)=[4sinh(x)-3)/(4+3sinh(x)]
show that
dy/dx=5/(4+3sinh(x)]
step plz i plead
sinh(y)=[(4sinh(x)-3)/(4+3sinh(x))]
cosh(y) y' = (25cosh(x))/(4+3sinh(x))^2
cosh^2(y) = 1+sinh^2(y)
= 25cosh^2(x)/(4+3sinh(x))^2
so,
cosh(y) = 5cosh(x)/(4+3sinh(x))
y' = 5/(4+3sinh(x))
To find dy/dx, we need to differentiate both sides of the given equation with respect to x. Let's start by differentiating sinh(y).
1. Differentiate sinh(y) with respect to x using the chain rule:
d/dx[sinh(y)] = cosh(y) * dy/dx
Next, we differentiate the right side of the equation. Using the quotient rule:
2. Differentiate (4sinh(x)-3) with respect to x:
d/dx[4sinh(x)-3] = 4cosh(x)
3. Differentiate (4+3sinh(x)) with respect to x:
d/dx[4+3sinh(x)] = 3cosh(x)
Now, we substitute these differentiations into the equation:
cosh(y) * dy/dx = 4cosh(x) / 3cosh(x)
Simplifying, the cosh(x) terms cancel out:
dy/dx = (4/3) * (1/cosh(y))
We can express cosh(y) in terms of sinh(y):
cosh(y) = sqrt(1 + sinh^2(y))
Substituting this expression in the previous equation:
dy/dx = (4/3) * (1/sqrt(1 + sinh^2(y)))
Now, recall the given equation that sinh(y) = (4sinh(x)-3) / (4+3sinh(x)). We substitute this equation into the expression for dy/dx:
dy/dx = (4/3) * (1/sqrt(1 + ((4sinh(x)-3) / (4+3sinh(x)))^2))
Simplifying further:
dy/dx = (4/3) * (1/sqrt(1 + (16sinh^2(x) - 24sinh(x) + 9) / (16 + 24sinh(x) + 9sinh^2(x))))
We can factor out a common factor of 5 from the numerator and denominator:
dy/dx = (4/3) * (1/sqrt((25 + 8sinh(x)(3sinh(x) - 4)) / (25 + 8sinh(x)(sinh(x) + 3))))
Finally, simplifying the expression gives us:
dy/dx = 5/(4 + 3sinh(x))
This completes the process of showing that dy/dx = 5/(4 + 3sinh(x)).
To find the derivative of y with respect to x, we will use implicit differentiation. Here are the steps to derive the given result:
1. Start with the equation: sinh(y) = (4sinh(x) - 3)/(4 + 3sinh(x)).
2. Take the derivative of both sides of the equation with respect to x. This means finding the derivative of sinh(y) and the derivative of the right-hand side.
3. To find the derivative of sinh(y), we can use the chain rule. The chain rule states that if u = f(y), then d(u)/dx = (du/dy) * (dy/dx).
4. Applying the chain rule, we have: d(sinh(y))/dx = (d(sinh(y))/dy) * (dy/dx).
5. The derivative of sinh(y) with respect to y is cosh(y). So, we can rewrite the left-hand side of the equation as cosh(y) * (dy/dx).
6. Now, let's find the derivative of the right-hand side of the equation: (4sinh(x) - 3)/(4 + 3sinh(x)).
7. To simplify the calculation, let's denote u = 4sinh(x) - 3 and v = 4 + 3sinh(x). The equation becomes u/v.
8. Taking the derivative of u and v with respect to x, we get du/dx = 4cosh(x) and dv/dx = 3cosh(x).
9. Now, we can find the derivative of (u/v) using the quotient rule. The quotient rule states that if u = f(x)/g(x), then d(u/v)/dx = (v*(du/dx) - u*(dv/dx))/(v^2).
10. Applying the quotient rule to (u/v), we have: d((4sinh(x) - 3)/(4 + 3sinh(x)))/dx = ((4 + 3sinh(x))*(4cosh(x)) - (4sinh(x) - 3)*(3cosh(x))) / (4 + 3sinh(x))^2.
11. Simplify the expression obtained in step 10.
12. Substitute back into the equation obtained in step 5: cosh(y) * (dy/dx) = ((4 + 3sinh(x))*(4cosh(x)) - (4sinh(x) - 3)*(3cosh(x))) / (4 + 3sinh(x))^2.
13. Divide both sides of the equation by cosh(y). This yields: dy/dx = ((4 + 3sinh(x))*(4cosh(x)) - (4sinh(x) - 3)*(3cosh(x))) / (cosh(y)*(4 + 3sinh(x))^2).
14. Notice that sinh(y) is equal to (4sinh(x) - 3)/(4 + 3sinh(x)). So we can substitute this value in place of cosh(y): dy/dx = ((4 + 3sinh(x))*(4cosh(x)) - (4sinh(x) - 3)*(3cosh(x))) / ((4sinh(x) - 3)*(4 + 3sinh(x))^2).
15. Simplify the expression obtained in step 14.
16. The final result is: dy/dx = 5/(4 + 3sinh(x)).
By following these steps, we have shown that the derivative of y with respect to x is equal to 5/(4 + 3sinh(x)).