derivative of cos^9(x^2y^6) — step-by-step solution

The expression

{\cos (x^{2} y^{6})}^{9}

contains several variables: the variable x is chosen for derivation

Let's perform the differentiation:

\frac{d}{d x} ({\cos (x^{2} y^{6})}^{9})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

\cos (x^{2} y^{6})

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({\cos (x^{2} y^{6})}^{9})

9 \cdot \frac{d}{d x} (\cos (x^{2} y^{6})) \cdot {\cos (x^{2} y^{6})}^{8}

9 \cdot \frac{d}{d x} (\cos (x^{2} y^{6})) \cdot {\cos (x^{2} y^{6})}^{8}

We differentiate using:

\frac{d}{d x} (\cos (f)) = - \frac{d}{d x} (f) \cdot \sin (f)

(where f =

x^{2} y^{6}

)

ⓘ The derivative of cos(x) is -sin(x):

\frac{d}{d x} (\cos (x^{2} y^{6}))

- \frac{d}{d x} (x^{2} y^{6}) \cdot \sin (x^{2} y^{6})

9 \cdot (- \frac{d}{d x} (x^{2} y^{6}) \cdot \sin (x^{2} y^{6})) \cdot {\cos (x^{2} y^{6})}^{8}

ⓘ The minus sign is placed in front of the product:

9 \cdot (- \frac{d}{d x} (x^{2} y^{6}))

-9 \cdot \frac{d}{d x} (x^{2} y^{6})

- 9 \cdot \frac{d}{d x} (x^{2} y^{6}) \cdot \sin (x^{2} y^{6}) \cdot {\cos (x^{2} y^{6})}^{8}

ⓘ For differentiation with respect to

x

y^{6}

is treated as constant. A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (x^{2} y^{6})

y^{6} \cdot \frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

- 9 \cdot y^{6} \cdot 2 x \cdot \sin (x^{2} y^{6}) \cdot {\cos (x^{2} y^{6})}^{8}

ⅰ

9 \cdot 2

18

- 18 y^{6} x \cdot \sin (x^{2} y^{6}) \cdot {\cos (x^{2} y^{6})}^{8}

▶Derivative:

- 18 x y^{6} \cdot {\cos (x^{2} y^{6})}^{8} \cdot \sin (x^{2} y^{6})