It's Still A Polynomial, Right?

Calculus Level 1

We know that $\dfrac d{dx} x^n = n x^{n-1}$ .

Is it also true that $\dfrac d{dx} n^x = x n^{x-1}$ ?

Yes, it is true No, it is not true

2 solutions

Chew-Seong Cheong
Jul 23, 2016

$\begin{aligned} \frac d{dx} n^x & = \frac d{dx} e^{x \ln n} = \ln n \ e^{x \ln n} = n^x \ln n \ne xn^{x-1} \end{aligned}$ $\boxed{\text{No, it is not true.}}$

Hung Woei Neoh
Jul 23, 2016

Let $y=n^x$

$\ln y=\ln n^x = x \ln n$

Use implicit differentiation:

$\dfrac{1}{y}\dfrac{dy}{dx}=\ln n\\ \dfrac{dy}{dx}=y\ln n = n^x \ln n \neq xn^{x-1}$

Therefore, $\boxed{\text{No, it is not true}}$

It's Still A Polynomial, Right?

2 solutions

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