Numerical Examples on Partial Differentiation
Example 1: First Order Partial Derivatives
Problem: If \( z = 3x^2y + 2xy^3 - 5y \), find \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
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Differentiate w.r.t \(x\):
\[ \frac{\partial z}{\partial x} = 6xy + 2y^3 \]Differentiate w.r.t \(y\):
\[ \frac{\partial z}{\partial y} = 3x^2 + 6xy^2 - 5 \]Example 2: Second Order Partial Derivatives
Problem: If \( z = x^3y^2 + 4xy \), find \( \frac{\partial^2 z}{\partial x^2} \).
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First derivative:
\[ \frac{\partial z}{\partial x} = 3x^2y^2 + 4y \]Second derivative:
\[ \frac{\partial^2 z}{\partial x^2} = 6xy^2 \]Example 3: Mixed Partial Derivatives
Problem: If \( z = x^2y^3 \), verify mixed partial derivatives.
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\[ \frac{\partial z}{\partial x} = 2xy^3 \] \[ \frac{\partial^2 z}{\partial y \partial x} = 6xy^2 \] \[ \frac{\partial z}{\partial y} = 3x^2y^2 \] \[ \frac{\partial^2 z}{\partial x \partial y} = 6xy^2 \]Hence verified.
Example 4: Chain Rule
Problem: If \( z = x^2 + y^2 \), where \( x = r\cos\theta \), \( y = r\sin\theta \), find \( \frac{\partial z}{\partial r} \).
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\[ z = r^2(\cos^2\theta + \sin^2\theta) = r^2 \] \[ \frac{\partial z}{\partial r} = 2r \]Example 5: Implicit Differentiation
Problem: If \( x^2 + y^2 + z^2 = 1 \), find \( \frac{\partial z}{\partial x} \).
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\[ 2x + 2z\frac{\partial z}{\partial x} = 0 \] \[ \frac{\partial z}{\partial x} = -\frac{x}{z} \]Example 6: Euler’s Theorem
Problem: Verify Euler’s theorem for \( z = x^2y^3 \).
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\[ \frac{\partial z}{\partial x} = 2xy^3 \] \[ \frac{\partial z}{\partial y} = 3x^2y^2 \] \[ x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = 5x^2y^3 = 5z \]Hence verified.
Example 7: Total Derivative
Problem: If \( z = x^2y + y^3 \), where \( x=t^2 \), \( y=t \), find \( \frac{dz}{dt} \).
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\[ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} \] \[ = (2xy)(2t) + (x^2+3y^2) \] Substitute \(x=t^2, y=t\): \[ \frac{dz}{dt} = 5t^4 + 3t^2 \]Example 8: Tangent Plane
Problem: Find tangent plane to \( z=x^2+y^2 \) at (1,1,2).
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\[ f_x=2x,\quad f_y=2y \] \[ z-2=2(x-1)+2(y-1) \] \[ z=2x+2y-2 \]Example 9: Maxima and Minima
Problem: Find stationary points of \( z=x^2+y^2-4x-6y \).
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\[ 2x-4=0 \Rightarrow x=2 \] \[ 2y-6=0 \Rightarrow y=3 \]Minimum at (2,3).
Example 10: Jacobian
Problem: If \( u=x+y \), \( v=x-y \), find \( \frac{\partial(u,v)}{\partial(x,y)} \).
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