derivative of y=19cos(x/b-2.5)+19

We save the definition:

y = 19 \cdot \cos (\frac{x}{b} - 2.5) + 19

The expression

19 \cdot \cos (\frac{x}{b} - 2.5) + 19

contains several variables: the variable x is chosen for differentiation

Let's differentiate:

\frac{d}{d x} (19 \cdot \cos (\frac{x}{b} - 2.5) + 19)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (19 \cdot \cos (\frac{x}{b} - 2.5) + 19)

\frac{d}{d x} (19 \cdot \cos (\frac{x}{b} - 2.5)) + 0

\frac{d}{d x} (19 \cdot \cos (\frac{x}{b} - 2.5)) + 0

Apply the differentiation rule:

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

with f =

\frac{x}{b} - 2.5

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (19 \cdot \cos (\frac{x}{b} - 2.5)) + 0

\frac{d}{d x} (19 \cdot \cos (\frac{x}{b} - 2.5))

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (19 \cdot \cos (\frac{x}{b} - 2.5))

19 \cdot \frac{d}{d x} (\cos (\frac{x}{b} - 2.5))

■The derivative of cos(x) is -sin(x):

\frac{d}{d x} (\cos (\frac{x}{b} - 2.5))

- \frac{d}{d x} (\frac{x}{b} - 2.5) \cdot \sin (\frac{x}{b} - 2.5)

19 \cdot (- \frac{d}{d x} (\frac{x}{b} - 2.5) \cdot \sin (\frac{x}{b} - 2.5))

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

19 \cdot (- \frac{d}{d x} (\frac{x}{b} - 2.5))

-19 \cdot \frac{d}{d x} (\frac{x}{b} - 2.5)

- 19 \cdot \frac{d}{d x} (\frac{x}{b} - 2.5) \cdot \sin (\frac{x}{b} - 2.5)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (\frac{x}{b} - 2.5)

\frac{d}{d x} (\frac{x}{b}) + 0

- 19 \cdot (\frac{d}{d x} (\frac{x}{b}) + 0) \cdot \sin (\frac{x}{b} - 2.5)

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (\frac{x}{b}) + 0

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

■A constant factor stays in front of the derivative, (k*u)' = k*u'. The derivative of x is 1:

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

\frac{1}{b}

- 19 \cdot \frac{1}{b} \cdot \sin (\frac{x}{b} - 2.5)

ⓘ Multiplying a fraction by a term multiplies its numerator:

- 19 \cdot \frac{1}{b} \cdot \sin (\frac{x}{b} - 2.5)

- \frac{19 \cdot 1 \cdot \sin (\frac{x}{b} - 2.5)}{b}

- \frac{19 \cdot \sin (\frac{x}{b} - 2.5)}{b}

▶Derivative:

- \frac{19 \cdot \sin (\frac{x}{b} - 2.5)}{b}