Derivatives of Exponentials
Read this topicExplanations 6
- I believe that we should compute ddxax\frac{d}{dx} a^xdxdax (i.e. with respect to x). To do so, let us define a constant eee equal to the limit of (1+1x)x\left(1 + \frac{1}{x}\…
By Kevin Ren, Ahaan Rungta
- Taking the limit definition of the derivative, ddx(ax)=limΔx→0ax+Δx−axΔx =limΔx→0ax(aΔx−1)Δx =axlimΔx→0aΔx−1Δx=axlimΔx→0a0+Δx−a0Δx=axddx(ax)∣x=0\begin{align*} \frac d{dx}(a^…
By Alexander Duan, Alex Federspiel
- Here is an interesting problem in elementary calculus:(taken from a problem solving group on Facebook.) f:[0,1]→Rf:[0,1]\rightarrow \textbf{R}f:[0,1]→R is continuous on [0,1][0,…
By Aritro Pathak
- (Video) Solving Every Derivative Ever by PatrickJMT Okay, maybe it's not EVERY derivative ever but here's a video by patrickJMT going over different derivatives for a whopping 3…
By Alex Federspiel
- (Video) Derivatives of Exponential Functions by PatrickJMT Here's a video by patrickJMT showing you how to find the derivative of exponential functions. The two main rules are: …
By Alex Federspiel
- (Videos) Derivatives of Exponential Functions by PatrickJMT Here's a video by PatrickJMT working through problems on finding the derivative of exponential functions. The two der…
By Alex Federspiel
Problems 24
- What is the derivative of 3x3^x3x?
- What is the derivative of xe−xxe^{-x}xe−x
- Calculate the following derivatives: i) 6x6^x6x ii) 6x+16^{x + 1}6x+1 NOTE: ii) can be done in two different ways. One is by using the Chain Rule. The other is to rewrite 6x+16^…
- Let aaa be the real number such that the curve y=exy = e^xy=ex is tangent to the curve y=ax2y = ax^2y=ax2 in the coordinate plane. Find ln(4a)\ln(4a)ln(4a). Source: HMMT Februa…
- Two differentiable real functions f,gf,gf,g satisfy f′(x)/g′(x)=ef(x)−g(x)f'(x)/g'(x) = e^{f(x) - g(x)}f′(x)/g′(x)=ef(x)−g(x) for all reals xxx, and f(0)=g(2003)=1f(0) = g(2003)…
- Calculate f′(x)f'(x)f′(x) if f(x)=e2xf(x)=e^{2x}f(x)=e2x.
- Calculate f′(x)f'(x)f′(x) if f(x)=esin(x2)f(x)=e^{\sin(x^2)}f(x)=esin(x2).
- What is the derivative of f(x)=xe+exf(x) = x^e + e^xf(x)=xe+ex?
- Differentiate eeexe^{e^{e^x}}eeex.
- What's the derivative of f(x)=axf(x) = a^xf(x)=ax?
- How do you take the derivative of axa^xax, where aaa is a constant?
- Find the derivative. f(x)=x3e3xf(x)=x^3e^{3x}f(x)=x3e3x
- Find the derivative. f(x)=exsinxf(x)=e^{x\sin x}f(x)=exsinx
- Find the derivative. f(x)=34x5f(x)=3^{4^{x^5}}f(x)=34x5
- If f(t)=3tf(t)=3^{t}f(t)=3t find a formula for f(n)(t)f^{(n)}(t)f(n)(t).
- Find the derivative. f(x)=e4xf(x)=e^{\sqrt{4^x}}f(x)=e4x
- Find the second derivative. f(x)=e2xsin(πx)f(x)=e^{2x}\sin(\pi x)f(x)=e2xsin(πx)
- Find the derivative. f(x)=e−csc−1xf(x)=e^{-\csc^{-1}x}f(x)=e−csc−1x
- Find the derivative. f(x)=arcsinesinxf(x)=\arcsin{e^{\sin x}}f(x)=arcsinesinx
- Find the derivative. y=2csc−1xsec−1xy=2^{\csc^{-1}x}\sec^{-1}xy=2csc−1xsec−1x
- Use differentials to estimate e0.07e^{0.07}e0.07.
- Use linearization to estimate e0.07e^{0.07}e0.07.
- Find the intervals on which the function f(x)=xex(x2ex)f(x)=xe^x(x^2e^x)f(x)=xex(x2ex) is increasing and decreasing.
- What can be said about the derivative of f(x)=(12)xf(x)=(\frac{1}{2})^xf(x)=(21)x