y = 6cos x - e^x, find dy/dx Step by Step

Step by step

Question: find dy/dx for y = 6 cos x − eˣ.

y = 6 cos x − e^x

The derivative of cos x is −sin x.

Mind the minus!

cos x → −sin x

eˣ is special: its derivative is itself.

Its slope always equals its height.

e^x → e^x

Use the rule on each term.

A number in front just waits, and multiplies.

dydx = 6 × (−sin x) − e^x

Careful!

The derivative of cos x is minus sin x.

Do not lose the minus.

Careful: not cos x → sin x ✗

So dy/dx = −6 sin x − eˣ.

= −6 sin x − e^x

Let's check: the derivative is the slope.

At x = 1, the rule gives −7.7671, with a calculator in radians.

dy/dx at x = 1: −6 sin 1 − e ≈ −7.7671

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about −7.77.

Very close to −7.7671.

Right!

y(1.001) − y(1)0.001 ≈ −7.77

Let's look back.

First: the rules.

Then: each term.

Then: tidy up.

Last: check.

Answer

= −6 sin x − e^x