What is the derivative of (1+2x)^x ?

The derivative of (1+2x)^x is (ln(1+2x)+(2x)/(1+2x))(1+2x)^x

What is the first derivative of (1+2x)^x ?

The first derivative of (1+2x)^x is (ln(1+2x)+(2x)/(1+2x))(1+2x)^x

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derivative of

(1 + 2 x)^{x}

(ln (1 + 2 x) + \frac{2 x}{1 + 2 x}) (1 + 2 x)^{x}

Solution steps

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

Apply exponent rule:

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

(1 + 2 x)^{x} = e^{x l n (1 + 2 x)}

= \frac{d}{d x} (e^{x l n (1 + 2 x)})

Apply the chain rule:

e^{x l n (1 + 2 x)} \frac{d}{d x} (x ln (1 + 2 x))

= e^{x l n (1 + 2 x)} \frac{d}{d x} (x ln (1 + 2 x))

\frac{d}{d x} (x ln (1 + 2 x)) = ln (1 + 2 x) + \frac{2 x}{1 + 2 x}

= e^{x l n (1 + 2 x)} (ln (1 + 2 x) + \frac{2 x}{1 + 2 x})

e^{x l n (1 + 2 x)} = (1 + 2 x)^{x}

= (ln (1 + 2 x) + \frac{2 x}{1 + 2 x}) (1 + 2 x)^{x}

What is the derivative of (1+2x)^x ?
The derivative of (1+2x)^x is (ln(1+2x)+(2x)/(1+2x))(1+2x)^x
What is the first derivative of (1+2x)^x ?
The first derivative of (1+2x)^x is (ln(1+2x)+(2x)/(1+2x))(1+2x)^x