What is the derivative of 5x ?

Question

Laura Worden · Accepted Answer

Explanation:  Given: ddx(5x)  Apply the general formula that ddx(ax)=axln⁡a  for  ∈R  ∴ddx(5x)=5xln⁡5

Maria Lopez · Accepted Answer

Explanation:  Let y=5x  differentiate implicity with respect to x  take ln⁡ (natural log) of both sides  ln⁡y=ln⁡5x=xln⁡5  ⇒1ydydx=ln⁡5  ⇒dydx=yln⁡5=5xln⁡5

nick1337 · Accepted Answer

For a generalized answer to this question, you can use the following which works in cases of $5^{x}$ and $x^{5}$ :
$\frac{d}{d x} (f (x)^{g (x)}) = f (x)^{g (x) - 1} (g (x) f^{'} (x) + f (x) \log (f (x)) g^{'} (x)$
So
$\frac{d}{d x} x^{5} = x^{5 - 1} (5 \times 1 + x \log (x) \times 0)$
$= x^{4} (5 + 0)$
$= 5 x^{4}$
And, likewise:
$\frac{d}{d x} 5^{x} = 5^{x - 1} (x \times 0 + 5 \log (5) \times)$
$= 5^{x - 1} (0 + 5 \log (5))$
$= 5^{x} \log (5)$
Note $5 \times 5^{x - 1} \equiv 5^{x}$

What is the derivative of 5^x ?

Answered question

Answer & Explanation

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