Partial Differentiation Practice

Calculus Level 2

If the function f f is given by

f ( x , y , z ) = x y + tan ⁡ ( x 2 y ) − z cos ⁡ ( x ) + log ⁡ ( x z ) , f(x,y,z) = xy + \tan\big(x^2 y\big) - z\cos(x) + \log(xz),

what is the partial derivative ∂ z f ? \partial_z f?

− cos ⁡ ( x ) + 1 z -\cos(x) + \frac{1}{z} − cos ⁡ ( x ) + x z -\cos(x) + \frac{x}{z} y + 2 x y sec ⁡ 2 ( x 2 y ) + z sin ⁡ ( x ) + 1 x y + 2xy \sec^2(x^2 y) + z\sin (x) + \frac{1}{x} − z cos ⁡ ( x ) + log ⁡ ( x z ) -z\cos(x) + \log(xz)

Matt DeCross by Matt DeCross , 27, USA

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1 solution

Matt DeCross
May 10, 2016

One ignores all functions not of z, treating all other variables as constant. This partial derivative is therefore just:

∂ z ( − z cos ⁡ x + log ⁡ ( x z ) ) = − cos ⁡ x + x 1 x z = − cos ⁡ x + 1 z . \partial_z (-z\cos x + \log (xz)) = -\cos x + x\frac{1}{xz} = -\cos x + \frac{1}{z}.

5 Helpful 1 Interesting 0 Brilliant 0 Confused

Does log(xz) represents logarithm for base 10 or e?

Danilo Popovic - 2 years, 4 months ago

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Base e. In higher level math, physics, etc base 10 is rarely used.

Matt DeCross - 2 years, 4 months ago

base for 10

huseyn rahimov - 1 year, 1 month ago

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