When differentiating an implicit equation, always ensure to apply the chain rule correctly. In cases where the equation involves trigonometric functions, be sure to use their derivative identities. Additionally, when isolating \(\frac{dy}{dx}\), carefully rearrange terms to simplify the expression. In this problem, substituting the expression for \(x\) into the denominator helped simplify the final result.
The given equation is:
\(\sin y = x \sin(a + y)\).
Differentiate both sides with respect to \(x\):
\(\cos y \frac{dy}{dx} = \sin(a + y) + x \cos(a + y) \frac{dy}{dx}\).
Rearrange to isolate \(\frac{dy}{dx}\):
\(\frac{dy}{dx} (\cos y - x \cos(a + y)) = \sin(a + y)\).
Simplify:
\(\frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cos(a + y)}\).
From the original equation \(\sin y = x \sin(a + y)\), rewrite \(x\) as:
\(x = \frac{\sin y}{\sin(a + y)}\).
Substitute \(x\) into the denominator:
\(\cos y - x \cos(a + y) = \cos y - \frac{\sin y \cos(a + y)}{\sin(a + y)}\).
Simplify the denominator:
\(\cos y - x \cos(a + y) = \frac{\cos y \sin(a + y) - \sin y \cos(a + y)}{\sin(a + y)} = \frac{\sin a}{\sin(a + y)}\).
Substitute this back into \(\frac{dy}{dx}\):
\(\frac{dy}{dx} = \frac{\sin(a + y)}{\frac{\sin a}{\sin(a + y)}} = \frac{\sin^2(a + y)}{\sin a}\).
Thus:
\(\boxed{\frac{\sin^2(a + y)}{\sin a}}\).
The given equation is:
\(\sin y = x \sin(a + y)\).
Differentiate both sides with respect to \(x\):
\[ \cos y \frac{dy}{dx} = \sin(a + y) + x \cos(a + y) \frac{dy}{dx}. \]Rearrange to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} (\cos y - x \cos(a + y)) = \sin(a + y). \]Simplify:
\[ \frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cos(a + y)}. \]From the original equation \(\sin y = x \sin(a + y)\), rewrite \(x\) as:
\[ x = \frac{\sin y}{\sin(a + y)}. \]Substitute \(x\) into the denominator:
\[ \cos y - x \cos(a + y) = \cos y - \frac{\sin y \cos(a + y)}{\sin(a + y)}. \]Simplify the denominator:
\[ \cos y - x \cos(a + y) = \frac{\cos y \sin(a + y) - \sin y \cos(a + y)}{\sin(a + y)} = \frac{\sin a}{\sin(a + y)}. \]Substitute this back into \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{\sin(a + y)}{\frac{\sin a}{\sin(a + y)}} = \frac{\sin^2(a + y)}{\sin a}. \]Thus:
\[ \boxed{\frac{\sin^2(a + y)}{\sin a}}. \]List-I (Function) | List-II (Derivative w.r.t. x) | |
---|---|---|
(A) \( \frac{5^x}{\ln 5} \) | (I) \(5^x (\ln 5)^2\) | |
(B) \(\ln 5\) | (II) \(5^x \ln 5\) | |
(C) \(5^x \ln 5\) | (III) \(5^x\) | |
(D) \(5^x\) | (IV) 0 |
List-I | List-II |
---|---|
The derivative of \( \log_e x \) with respect to \( \frac{1}{x} \) at \( x = 5 \) is | (I) -5 |
If \( x^3 + x^2y + xy^2 - 21x = 0 \), then \( \frac{dy}{dx} \) at \( (1, 1) \) is | (II) -6 |
If \( f(x) = x^3 \log_e \frac{1}{x} \), then \( f'(1) + f''(1) \) is | (III) 5 |
If \( y = f(x^2) \) and \( f'(x) = e^{\sqrt{x}} \), then \( \frac{dy}{dx} \) at \( x = 0 \) is | (IV) 0 |
List-I (Name of account to be debited or credited, when shares are forfeited) | List-II (Amount to be debited or credited) |
---|---|
(A) Share Capital Account | (I) Debited with amount not received |
(B) Share Forfeited Account | (II) Credited with amount not received |
(C) Calls-in-arrears Account | (III) Credited with amount received towards share capital |
(D) Securities Premium Account | (IV) Debited with amount called up |