What is the derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) ?

The derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) is

What is the first derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) ?

The first derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) is

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\frac{d}{d x} ((2 r (x) a^{r (x) x} + n)^{p (x)})

(ln (2 a^{x r (x)} r (x) + n) \frac{d}{d x} (p (x)) + \frac{2 ( a ^{x r (x)} \frac{d}{d x} ( r ( x ) ) + a ^{x r (x)} ln ( a ) r ( x ) ( x \frac{d}{d x} ( r ( x ) ) + r ( x ) ) ) p ( x )}{2 a ^{x r (x)} r ( x ) + n}) (2 a^{x r (x)} r (x) + n)^{p (x)}

Solution steps

\frac{d}{d x} ((2 r (x) a^{r (x) x} + n)^{p (x)})

Apply exponent rule:

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

(2 r (x) a^{r (x) x} + n)^{p (x)} = e^{p (x) l n (2 r (x) a^{r (x) x} + n)}

= \frac{d}{d x} (e^{p (x) l n (2 r (x) a^{r (x) x} + n)})

Apply the chain rule:

e^{p (x) l n (2 r (x) a^{r (x) x} + n)} \frac{d}{d x} (p (x) ln (2 r (x) a^{r (x) x} + n))

= e^{p (x) l n (2 r (x) a^{r (x) x} + n)} \frac{d}{d x} (p (x) ln (2 r (x) a^{r (x) x} + n))

\frac{d}{d x} (p (x) ln (2 r (x) a^{r (x) x} + n)) = ln (2 a^{x r (x)} r (x) + n) \frac{d}{d x} (p (x)) + \frac{2 ( a ^{x r (x)} \frac{d}{d x} ( r ( x ) ) + a ^{x r (x)} ln ( a ) r ( x ) ( x \frac{d}{d x} ( r ( x ) ) + r ( x ) ) ) p ( x )}{2 a ^{x r (x)} r ( x ) + n}

= e^{p (x) l n (2 r (x) a^{r (x) x} + n)} (ln (2 a^{x r (x)} r (x) + n) \frac{d}{d x} (p (x)) + \frac{2 ( a ^{x r (x)} \frac{d}{d x} ( r ( x ) ) + a ^{x r (x)} ln ( a ) r ( x ) ( x \frac{d}{d x} ( r ( x ) ) + r ( x ) ) ) p ( x )}{2 a ^{x r (x)} r ( x ) + n})

= e^{p (x) l n (2 r (x) a^{r (x) x} + n)} (ln (2 a^{x r (x)} r (x) + n) \frac{d}{d x} (p (x)) + \frac{2 ( a ^{x r (x)} \frac{d}{d x} ( r ( x ) ) + a ^{x r (x)} ln ( a ) r ( x ) ( x \frac{d}{d x} ( r ( x ) ) + r ( x ) ) ) p ( x )}{2 a ^{x r (x)} r ( x ) + n})

e^{p (x) l n (2 r (x) a^{r (x) x} + n)} = (2 a^{x r (x)} r (x) + n)^{p (x)}

= (ln (2 a^{x r (x)} r (x) + n) \frac{d}{d x} (p (x)) + \frac{2 ( a ^{x r (x)} \frac{d}{d x} ( r ( x ) ) + a ^{x r (x)} ln ( a ) r ( x ) ( x \frac{d}{d x} ( r ( x ) ) + r ( x ) ) ) p ( x )}{2 a ^{x r (x)} r ( x ) + n}) (2 a^{x r (x)} r (x) + n)^{p (x)}

y^{^{'}} = \frac{( - y ^{2} )}{- ( x y - x ^{3} )}

\frac{\partial}{\partial y} (7 sin (x y))

\frac{\partial}{\partial x} (3 x e^{5 x y})

x \to - 4 lim (x)

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

What is the derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) ?
The derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) is
What is the first derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) ?
The first derivative of (2{r}(xa^{{r}(x)x}+n)^{{p}(x)}) is