What is the derivative of cos^{ln(7x}(x)) ?

The derivative of cos^{ln(7x}(x)) is ((ln(cos(x)))/x-tan(x)ln(7x))(7x)^{ln(cos(x))}

What is the first derivative of cos^{ln(7x}(x)) ?

The first derivative of cos^{ln(7x}(x)) is ((ln(cos(x)))/x-tan(x)ln(7x))(7x)^{ln(cos(x))}

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

(\frac{ln ( cos ( x ) )}{x} - tan (x) ln (7 x)) (7 x)^{l n (c o s (x))}

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

cos^{l n (7 x)} (x) = e^{l n (7 x) l n (c o s (x))}

= \frac{d}{d x} (e^{l n (7 x) l n (c o s (x))})

Apply the chain rule:

e^{l n (7 x) l n (c o s (x))} \frac{d}{d x} (ln (7 x) ln (cos (x)))

= e^{l n (7 x) l n (c o s (x))} \frac{d}{d x} (ln (7 x) ln (cos (x)))

\frac{d}{d x} (ln (7 x) ln (cos (x))) = \frac{ln ( cos ( x ) )}{x} - tan (x) ln (7 x)

= e^{l n (7 x) l n (c o s (x))} (\frac{ln ( cos ( x ) )}{x} - tan (x) ln (7 x))

e^{l n (7 x) l n (c o s (x))} = (7 x)^{l n (c o s (x))}

= (\frac{ln ( cos ( x ) )}{x} - tan (x) ln (7 x)) (7 x)^{l n (c o s (x))}

What is the derivative of cos^{ln(7x}(x)) ?
The derivative of cos^{ln(7x}(x)) is ((ln(cos(x)))/x-tan(x)ln(7x))(7x)^{ln(cos(x))}
What is the first derivative of cos^{ln(7x}(x)) ?
The first derivative of cos^{ln(7x}(x)) is ((ln(cos(x)))/x-tan(x)ln(7x))(7x)^{ln(cos(x))}