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### Course: Differential Calculus (2017 edition)>Unit 6

Lesson 6: Sine & cosine derivatives

# Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)

Learn the derivatives of several common functions. Created by Sal Khan.

## Want to join the conversation?

• Lets say I have an equation sin x = 1/2. Then clearly, x = 30 degrees or pi/6 radians.
Now if I differentiate both sides of the equation with respect to x (because both are equal, their derivatives should also be equal), I will have cos x = 0, right?
But that means x = 90 degrees, which is obviously not the solution! Could somebody please explain why this is happening?
• Once you say that "x = some number", x is no longer a variable but rather it is a constant -- and you must then treat it like a constant.

Since x = π/6, both "x" and "sin(x)" are constants (because sin(π/6) = 1/2) and the derivative of a constant is zero.

Does this clear things up for you?
• What is e?
• Think of e like pi - it's just an accepted name for a number that's useful in a bunch of formulas. It's equal to around 2.71828, so plug that into wherever you see e and you'll be good, or you can often leave things in terms of e.
• What is the explanation of why the derivative of sec (x) is sec (x) tan(x)?

It comes up in the questions for this topic on Khan Academy in the Special derivatives exercise.
• If we accept that d/dx (cos x) = − sin x, and the power rule then:
sec x ≡ 1/cos x
Let u = cos x, thus du = − sin x dx
sec x = 1/u
(1/u) = (u⁻¹)
By the power rule:
derivative of (u⁻¹) = −u⁻² du
Back substituting:
= −(cos x)⁻² ( − sin x) ∙ dx
= [sin x / (cos x)²] ∙ dx
= [(sin x / cos x) ∙ (1/cos x)] ∙ dx
= [tan (x) ∙ sec (x)] ∙ dx
• I think I understand how to do the derivatives of the trig functions. But What if instead of Sin(x,
there was Sin( and equation. Like, Sin(x^2+2). How would you take the derivative when it's something like that, and not just Sin or Cos?
• You always have to multiply the outer derivative with the inner derivative. That's true even for sin(x), it's just that the inner derivative is 1. (d/dx x = 1)

d/dx sin(x) = cos(x) * 1 = cos(x)

d/dx sin(2x) = cos(2x) * 2 = 2 cos(2x)

d/dx sin(x^2) = cos(x^2) * 2x = 2x cos(x^2)

d/dx sin(x^2 + 2) = cos(x^2 + 2) * 2x = 2x cos(x^2 + 2)
• Where i can find the proof of these derivatives?
• The website https://proofwiki.org has proofs for many mathematical identities and theorems, including various common derivatives.
• Is there a way of proving the derivative of e^x without using the derivative of ln(x)? If yes, can you please provide a link to it?
• Yes, I can prove it:
derivative of e^x , by definition of derivative:
= lim h→0 { e^(x+h) - e^x} / h
= lim h → 0 {e^x(e^h) -e^x}/h
= lim h → 0 e^x{(e^h) -1}/h
= (e^x) lim h → 0 {(e^h) -1}/h
By definition of e:
e = lim h→ 0 (1+h)^(1/h)
And so,
e^h = lim h→ 0 (1+h)^(h/h)
= lim h→ 0 (1+h)
Substituting that in:
= (e^x) lim h → 0 {(1+h) -1}/h
= (e^x) lim h → 0 { h/h}
= (e^x) lim h → 0 { 1 }
= e^x * (1)
= e^x
• What are the proofs of sin, cos & tan derivatives? Are there any videos proving them?
• Is there a video that goes over the intuition behind integrals of trig functions (sec(x) and cot(x) included ideally)?
• Are there videos of proving the derivations of trig functions? I've only seen the sine limit approaching zero proof where he proves it using sandwhich / squeeze theorem, (that is proof that limit of sinx/x is equal to 1 as x approaches zero). Also, could I use squeeze function and 2 parabolas to prove sine function instead of the overly complicated way he's proving it? And another thing, how can derivative of function tanx be cosecant when he's clearly written 1/cos^2(x) which is cosecant squared?
(1 vote)
• Unfortunately there's no proof currently on Khan of the derivatives of sine, cosine, or tangent.
Also, the derivative of tangent is secant squared.
`1/cos x = sec x`
`d/dx (tan x) = 1/cos^2 x = sec^2 x`

As for proofs, here's a good proof of the derivative of sine:
https://proofwiki.org/wiki/Derivative_of_Sine_Function/Proof_2

Using the proof for sine, you can easily prove cosine using the equality
`cos(x) = sin(x+π/2)` and the chain rule.

Using the derivative of sine and the derivative of cosine, you can use the definition of tangent
`tan(x) = sin(x)/cos(x)` and the quotient rule to prove the derivative of tangent.