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### Course: AP®︎/College Calculus BC>Unit 2

Lesson 8: Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)

# Derivative of 𝑒ˣ

The derivative of 𝑒ˣ is... well... 𝑒ˣ. This is a very special property lies at the heart of our work with exponential functions.

## Want to join the conversation?

• How would you use the "lim h-->0 f(x+h) - f(x)/h" strategy to solve for the derivative of e^x? I got 0/0, but it doesn't look like factoring or using conjugates work.
• Factoring will work!

f(x)=e^x : this will be our original equation that we want to differentiate to achieve the general formula. As noted by this video, the general formula for this equation is the equation itself: e^x. Let's prove it using the general limit notation!

First, plug in (x) and (x+h) into the exponent.
f(x)= e^x
f(x+h)=e^(x+h)

Now that we have established the terms for the limit notation, we can continue:
lim h-->0 (e^(x+h) - e^x)/h
lim h-->0 (e^x times e^h - e^x)/h : bare with me, I know it's difficult to illustrate this but remind yourself that a^(b+c) is equivalent to a^b times a^c, as long as the base is the same, according the the exponential laws. Thus, e^(x+h) can be separated as e^x times e^h, since multiplying the two bases with the same exponent simply means to add the exponents together.

lim h-->0 [e^x(e^h - 1)]/h : factor out e^x, since this is the common factor. If you did not recognize this operation, you can set e^x = a
*** lim h-->0 (a times e^h - a)/h
*** lim h--->0 [a (e^h - 1)]/h : where e^x = a

lim h-->0 [e^x(e^h - 1)]/h : the coefficient, e^x does not depend on the denominator; it can be interpreted as (e^x)/1 times (e^h - 1)/h. Thus, move the e^x in front of the lim.

e^x times lim h-->0 (e^h - 1)/h : approximate the value of the limit by picking a number very close to zero to input into 'h' value. Let's choose .0001 (you can choose .01, .001, .0000000000000000000001...but you will find that as h gets closer to zero, the value of the limit gets more accurate).

e^x times lim h-->0 (e^0.0001 - 1)/0.0001 : the value of the limit is 1
e^x times 1
f'(x)= e^ x : this proves that the derivative (general slope formula) of f(x)= e^x is e^x, which is the function itself. In other words, for every point on the graph of f(x)=e^x, the slope of the tangent is equal to the y-value of tangent point. So if y= 2, slope will be 2. if y= 2.12345, slope will be 2.12345
• Wouldn't e^x fall under the power rule? So why does it behave differently?
• The power rule concerns functions of the form x^n, where the base is the variable and the exponent is a constant.

In the case of e^x, the base is constant and the exponent is variable, so it isn't the same thing.
• But shouldn't be the derivative of e^x = xe^(x-1) using Power Rule?
• No the power rule only works for a constant in the exponent, when the exponent is the input to the function or some function of the input to the function, you cannot use the power rule. Hope this helps!
• How did he got the slope so fast??
• He's just eyeballing it; it looks pretty close to the values he says.
• At , Sal says "the derivative with respect to X". Can we have a derivative in respect to another variable, say, t (time) for a function of time (f(t)) like speed in a race. Would the derivative be expressed as d/dt f(t)? Is this possible?
• You are exactly right. You can take the derivative with respect to any variable and it will look just like that.

Also, like you said, the variable should match the function. If you wanted to take d/dx f(t), it would be 0, because f(t) does not depend on x or change as x changes.
• Does that mean, at y = infinity, the graph will be a vertical line and thus, the slope of e^x at y = infinity will be undefined?
• yuh, i guess if ur talking about y=infinity that's the case. The slope for that equation will approach infinity as x->oo. same thing happens with equations like x^2
(1 vote)
• What would the derivative of e^(-5x) be then? Would it be the same thing?
(1 vote)
• You apply the chain rule, first saying e^(-5x) = e^u*, where u = -5x and du/dx = -5. Differentiating, you get du/dx e^u, which equals -5e^(-5x).
• Is this function, e^(x) really differentiable?if yes then how. My math teacher says it isn't actually differentiable.
• Yes, the function e^x is differentiable, and its derivative is e^x.

Have a blessed, wonderful day!