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

The derivative of x^{{r}(x}) is x^{{r}(x)}(ln(x) derivative of {r}(x)+({r}(x))/x)

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

The first derivative of x^{{r}(x}) is x^{{r}(x)}(ln(x) derivative of {r}(x)+({r}(x))/x)

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

x^{r (x)} (ln (x) \frac{d}{d x} (r (x)) + \frac{r ( x )}{x})

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (r (x) ln (x)) = ln (x) \frac{d}{d x} (r (x)) + \frac{r ( x )}{x}

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

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

= x^{r (x)} (ln (x) \frac{d}{d x} (r (x)) + \frac{r ( x )}{x})

\frac{d}{d x} (30)

- x^{2} + 3 x, x - 3

\frac{d y}{d x} = \frac{( ( 1 + y ) ^{2} )}{( 1 + x ) ^{3}}

\int e^{- 2 x} cos (e^{- x}) d x

\frac{d}{d y} (\frac{x + y}{x - y})

What is the derivative of x^{{r}(x}) ?
The derivative of x^{{r}(x}) is x^{{r}(x)}(ln(x) derivative of {r}(x)+({r}(x))/x)
What is the first derivative of x^{{r}(x}) ?
The first derivative of x^{{r}(x}) is x^{{r}(x)}(ln(x) derivative of {r}(x)+({r}(x))/x)