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

The derivative of 5x^{ln(x}) is (10e^{ln^2(x)}ln(x))/x

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

The first derivative of 5x^{ln(x}) is (10e^{ln^2(x)}ln(x))/x

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

\frac{10 e ^{l n^{2} (x)} ln ( x )}{x}

Solution steps

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

Take the constant out:

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

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

5 e^{l n (x) l n (x)} \frac{2 ln ( x )}{x} : \frac{10 e ^{l n^{2} (x)} ln ( x )}{x}

= \frac{10 e ^{l n^{2} (x)} ln ( x )}{x}

\frac{d y}{d x} y = x^{x}

8 x^{3} - 18 x^{2} - 6

\int - 2 x + 18 x^{2} - 4 x^{3} + 16 d x

x \to 3 lim (4 x^{2} - 2 x - 6)

y = \frac{x}{x + 9}, (1, 0.1)

What is the derivative of 5x^{ln(x}) ?
The derivative of 5x^{ln(x}) is (10e^{ln^2(x)}ln(x))/x
What is the first derivative of 5x^{ln(x}) ?
The first derivative of 5x^{ln(x}) is (10e^{ln^2(x)}ln(x))/x