y = 2^(5x), find dy/dx Step by Step

Step by step

Question: find dy/dx for y = 2^(5x).

y = 2^(5x)

We know how to take the derivative of e to a power.

And any number is e to a power: 2 is e to the ln 2.

2 = e^(ln 2)

So 2 to the 5x is e to the 5x times ln 2.

2^(5x) = e^((5x) ln 2)

The derivative of e to a power is itself, times the slope of the power: 5 times ln 2.

ddxe^((5x) ln 2) = e^((5x) ln 2) × 5 × ln 2

The rule: b to the x gives b to the x, times ln b.

bˣ → bˣ ln b

Careful!

The power rule is only for x to a fixed number.

Here x is in the power: use b to the x, times ln b.

Careful: not 2ˣ → x2^(x − 1) ✗

Put it back in base 2, with the numbers in front: dy/dx = 5 × 2^(5x) ln 2.

dydx = 5 × 2^(5x) ln 2

Let's check: the derivative is the slope.

At x = 0, our answer gives 3.4657, with a calculator.

dy/dx at x = 0: 3.4657

Now the slope between two very close points: x = 0 and x = 0.001.

A calculator gives about 3.47.

Very close to 3.4657.

Right!

y(0.001) − y(0)0.001 ≈ 3.47

Let's look back.

First: write it with e.

Then: the rule.

Then: tidy up.

Last: check.

Answer

dydx = 5 × 2^(5x) ln 2