What is the derivative of 5x^{ln(5x}) ?

The derivative of 5x^{ln(5x}) is 5^{1+ln(x)}x^{ln(x)-1}ln(5x^2)

What is the first derivative of 5x^{ln(5x}) ?

The first derivative of 5x^{ln(5x}) is 5^{1+ln(x)}x^{ln(x)-1}ln(5x^2)

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\frac{d}{d x} (5 x^{l n (5 x)})

5^{1 + l n (x)} x^{l n (x) - 1} ln (5 x^{2})

Solution steps

\frac{d}{d x} (5 x^{l n (5 x)})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{d}{d x} (x^{l n (5 x)})

Apply exponent rule:

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

x^{l n (5 x)} = e^{l n (5 x) l n (x)}

= 5 \frac{d}{d x} (e^{l n (5 x) l n (x)})

Apply the chain rule:

e^{l n (5 x) l n (x)} \frac{d}{d x} (ln (5 x) ln (x))

= e^{l n (5 x) l n (x)} \frac{d}{d x} (ln (5 x) ln (x))

\frac{d}{d x} (ln (5 x) ln (x)) = \frac{ln ( 5 x ^{2} )}{x}

= 5 e^{l n (5 x) l n (x)} \frac{ln ( 5 x ^{2} )}{x}

Simplify

5 e^{l n (5 x) l n (x)} \frac{ln ( 5 x ^{2} )}{x} : 5^{1 + l n (x)} x^{l n (x) - 1} ln (5 x^{2})

= 5^{1 + l n (x)} x^{l n (x) - 1} ln (5 x^{2})

y^{^{'}} + 2 x y = \frac{x}{y ^{2}}

\int (3 x^{2} - 2 x + 1) (x^{3} - x^{2} + x)^{8} d x

y = (t \cdot sin (t))^{2} + t \cdot sin (t)

sec^{2} (2 x) - tan^{2} (2 x)

\frac{\partial}{\partial x} (- 4 e^{- x - y})

What is the derivative of 5x^{ln(5x}) ?
The derivative of 5x^{ln(5x}) is 5^{1+ln(x)}x^{ln(x)-1}ln(5x^2)
What is the first derivative of 5x^{ln(5x}) ?
The first derivative of 5x^{ln(5x}) is 5^{1+ln(x)}x^{ln(x)-1}ln(5x^2)