What is the derivative of x^r ?

The derivative of x^r is x^r(ln(x)(dr)/(dx)+r/x)

What is the first derivative of x^r ?

The first derivative of x^r is x^r(ln(x)(dr)/(dx)+r/x)

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

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