Question

# Find the first partial derivatives of the following functions. g(x,y)=\cos^{5}(x^{2}y^{3})

Derivatives
Find the first partial derivatives of the following functions.
$$\displaystyle{g{{\left({x},{y}\right)}}}={{\cos}^{{{5}}}{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}$$

2021-03-28
Step 1
To find the partial derivatives.
$$\displaystyle{g{{\left({x},{y}\right)}}}={{\cos}^{{{5}}}{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}$$
Step 2
Let us first evaluate the partial derivative with respect to x.
We differentiate the function with respect to x keeping y constant.
$$\displaystyle{\frac{{\partial{g{{\left({x},{y}\right)}}}}}{{\partial{x}}}}={5}{{\cos}^{{{4}}}{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}{\left(-{\sin{{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}}\right)}{\left({2}{x}{y}^{{{3}}}\right)}$$ (Chain Rule of differentiation is also applicable)
$$\displaystyle=-{10}{x}{y}^{{{3}}}{{\cos}^{{{4}}}{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}{\sin{{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}}$$
Step 3
Now, let us evaluate the partial derivative with respect to y.
We differentiate the function with respect to y keeping x constant.
$$\displaystyle{\frac{{\partial{g{{\left({x},{y}\right)}}}}}{{\partial{y}}}}={5}{{\cos}^{{{4}}}{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}{\left(-{\sin{{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}}\right)}{\left({3}{x}^{{{2}}}{y}^{{{2}}}\right)}$$ (Chain Rule of differentiation is also applicable)
$$\displaystyle=-{15}{x}^{{{2}}}{y}^{{{2}}}{{\cos}^{{{4}}}{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}{\sin{{\left({x}^{{{2}}}{y}^{{{3}}}\right)}}}$$