Calculate all partial derivatives of the function g(x,y,z)=xsin⁡(y)z2

Question

Calculate all partial derivatives of the function  g(x,y,z)=xsin⁡(y)z2

Mike Henson · Accepted Answer

Step 1  The given function is:  g(x,y,z)=xsin⁡(y)z2  Step 2  Now the partial derivation of the function with respect to x is:  ∂g∂x=∂∂x(xsin⁡(y)z2)  ∂g∂x=sin⁡(y)z2  Now the partial derivation of the function with respect to y is:  ∂g∂y=∂∂y(xsin⁡(y)z2)  ∂g∂y=xz2∂∂y(sin⁡y)  ∂g∂y=xcos⁡(y)z2  Now the partial derivation of the function with respect to z is:  ∂g∂z=∂∂z(xsin⁡(y)z2)  ∂g∂z=xsin⁡(y)∂∂z(1z2)  ∂g∂z=−xsin⁡(y)z

Angsts · Accepted Answer

Step 1  Given:  g(x,y,z)=xsin⁡(y)z2  Step 2  ∂∂x(xsin⁡(y)z2)  Treat y, z as constants  Take the constant out: (a*f)=a*f  =sin⁡(y)z2∂∂x(x)  Apply the common derivative: ∂∂x(x)=1  =sin⁡(y)z2⋅1  Simplify  =sin⁡(y)z2  Step 3  ∂∂y(xsin⁡(y)z2)  Treat x, z as constants  Take the constant out: (a*f)=a*f  =xz2∂∂y(sin⁡(y))  Apply the common derivative: ∂∂y(sin⁡(y))=cos⁡(y)  =xz2cos⁡(y)  Multiply fractions: a⋅bc=a⋅bc  =xcos⁡(y)z2  Step 4  ∂∂z(xsin⁡(y)z2)  Treat x, y as constants  Take the constant out: (a*f)=a*f  =xsin⁡(y)∂∂z(1z2)  Apply exponent rule: 1a=a−1  =xsin⁡(y)∂∂z(z−2)  Apply the Power Rule: ddx(xa)=a⋅xa−1  =xsin⁡(y)(−2z−2−1)  Simplify xsin⁡(y)(−2z−2−1):−2xsin⁡(y)z3  =−2xsin⁡(y)

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Calculate all partial derivatives of the function g(x,y,z)=\frac{x\sin (y

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