derivative of f(x)=x^9cos(x) — step-by-step solution

We save the definition:

f (x) = x^{9} \cdot \cos (x)

1) Let's differentiate:

\frac{d}{d x} (x^{9} \cdot \cos (x))

For

u1

x^{9}

and

v2

\cos (x)

, use the differentiation rule:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's differentiate

u1

\frac{d}{d x} (x^{9})

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{9})

9 x^{8}

9 x^{8}

3) Let's differentiate

v2

\frac{d}{d x} (\cos (x))

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

\frac{d}{d x} (\cos (x))

- \sin (x)

- \sin (x)

4) We evaluate the resulting derivative (

u1 \cdot v2

)':

9 x^{8} \cdot \cos (x) + x^{9} \cdot (- \sin (x))

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

x^{9} \cdot (- \sin (x))

- x^{9} \cdot \sin (x)

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

5) Then you can resume the differentiation:

ⅰ

\frac{d}{d x} (x^{9} \cdot \cos (x))

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

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

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

▶Derivative:

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