derivative of (cos(7x)^x) — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} ({\cos (7 x)}^{x})

To differentiate on intervals where

\cos (7 x) > 0

, write

{\cos (7 x)}^{x}

e^{x \cdot \ln (\cos (7 x))}

ⅰ

\frac{d}{d x} ({\cos (7 x)}^{x})

\frac{d}{d x} (x \cdot \ln (\cos (7 x))) \cdot e^{x \cdot \ln (\cos (7 x))}

\frac{d}{d x} (x \cdot \ln (\cos (7 x))) \cdot e^{x \cdot \ln (\cos (7 x))}

We differentiate

u1

x

and

v2

\ln (\cos (7 x))

using:

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

2) Let's perform the differentiation

u1

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

ⓘ The derivative of x is 1:

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

1

1

3) Let's perform the differentiation

v2

\frac{d}{d x} (\ln (\cos (7 x)))

We differentiate using:

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

(where f =

\cos (7 x)

)

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (\cos (7 x)))

\frac{d}{d x} (\cos (7 x)) \cdot \frac{1}{\cos (7 x)}

\frac{d}{d x} (\cos (7 x)) \cdot \frac{1}{\cos (7 x)}

We differentiate using:

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

(where f =

7 x

)

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (\cos (7 x)) \cdot \frac{1}{\cos (7 x)}

\frac{\frac{d}{d x} (\cos (7 x)) \cdot 1}{\cos (7 x)}

\frac{\frac{d}{d x} (\cos (7 x))}{\cos (7 x)}

■The derivative of cos(x) is -sin(x):

\frac{d}{d x} (\cos (7 x))

- \frac{d}{d x} (7 x) \cdot \sin (7 x)

- \frac{\frac{d}{d x} (7 x) \cdot \sin (7 x)}{\cos (7 x)}

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

\frac{d}{d x} (7 x)

7

- \frac{7 \cdot \sin (7 x)}{\cos (7 x)}

ⓘ sin(x)/cos(x) = tan(x):

- \frac{7 \cdot \sin (7 x)}{\cos (7 x)}

- 7 \cdot \tan (7 x)

- 7 \cdot \tan (7 x)

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

1 \cdot \ln (\cos (7 x)) + x \cdot (- 7 \cdot \tan (7 x))

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

x \cdot (-7) \cdot \tan (7 x)

- 7 x \cdot \tan (7 x)

\ln (\cos (7 x)) - 7 x \cdot \tan (7 x)

5) Then you can continue the derivation:

ⅰ

\frac{d}{d x} (x \cdot \ln (\cos (7 x)))

\ln (\cos (7 x)) - 7 x \cdot \tan (7 x)

(\ln (\cos (7 x)) - 7 x \cdot \tan (7 x)) \cdot {\cos (7 x)}^{x}

ⓘ Expand by multiplying each term inside the parentheses:

(\ln (\cos (7 x)) - 7 x \cdot \tan (7 x)) \cdot {\cos (7 x)}^{x}

\ln (\cos (7 x)) \cdot {\cos (7 x)}^{x} - 7 x \cdot \tan (7 x) \cdot {\cos (7 x)}^{x}

\ln (\cos (7 x)) \cdot {\cos (7 x)}^{x} - 7 x \cdot \tan (7 x) \cdot {\cos (7 x)}^{x}

{\cos (7 x)}^{x} \cdot \ln (\cos (7 x)) - 7 x \cdot {\cos (7 x)}^{x} \cdot \tan (7 x)

{\cos (7 x)}^{x} \cdot (\ln (\cos (7 x)) - 7 x \cdot \tan (7 x))

▶Derivative:

{\cos (7 x)}^{x} \cdot (\ln (\cos (7 x)) - 7 x \cdot \tan (7 x))