derivative of 6ln(cos(x)) — step-by-step solution

Let's perform the differentiation:

\frac{d}{d x} (6 \cdot \ln (\cos (x)))

Apply the differentiation rule:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

with f =

\cos (x)

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

\frac{d}{d x} (6 \cdot \ln (\cos (x)))

6 \cdot \frac{d}{d x} (\ln (\cos (x)))

■The derivative of ln(x) is 1/x:

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

\frac{d}{d x} (\cos (x)) \cdot \frac{1}{\cos (x)}

6 \cdot \frac{d}{d x} (\cos (x)) \cdot \frac{1}{\cos (x)}

ⓘ Multiplying a fraction by a term multiplies its numerator:

6 \cdot \frac{d}{d x} (\cos (x)) \cdot \frac{1}{\cos (x)}

\frac{6 \cdot \frac{d}{d x} (\cos (x)) \cdot 1}{\cos (x)}

\frac{6 \cdot \frac{d}{d x} (\cos (x))}{\cos (x)}

■The derivative of cos(x) is -sin(x):

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

- \sin (x)

\frac{6 \cdot (- \sin (x))}{\cos (x)}

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

6 \cdot (- \sin (x))

- 6 \cdot \sin (x)

- \frac{6 \cdot \sin (x)}{\cos (x)}

ⓘ sin(x)/cos(x) = tan(x):

- \frac{6 \cdot \sin (x)}{\cos (x)}

- 6 \cdot \tan (x)

▶Derivative:

- 6 \cdot \tan (x)