Chain Rule
Read this topicExplanations 18
- See Pete Clark's physical intuition here, at point (2). For a proof using linear approximation and (implicitly) little-o notation, see this proof on Wikipedia. (In fact, see if …
By Victor Wang, Moor Xu
- General formula for functions f(x)f(x)f(x) and g(x)g(x)g(x): ddx(f(g(x))=f′(g(x))g′(x)\frac{d}{dx}(f(g(x)) = f'(g(x))g' (x)dxd(f(g(x))=f′(g(x))g′(x) Solution: Since the derivat…
By Joshua Siktar, Leigh Marie Braswell, Victor Wang, Alex Federspiel
- Let's try to figure out the derivative of (x+2)2(x+2)^2(x+2)2. This is actually the composition of two functions! Let: f(x)=x+2f(x) = x+2f(x)=x+2 and g(x)=x2.g(x) = x^2.g(x)=x2.…
By Po-Shen Loh
- Suppose you were asked to differentiate the function f(x)=x2+1.f(x) = \sqrt{x^2 + 1}.f(x)=x2+1. The differentiation formulas such as the Product Rule and Quotient Rule are not …
By Daniel Whatley, Andrew Machkasov, Moor Xu, Victor Wang
- Chain Rule from Bicycles Let's start with something straightforward, which doesn't even involve Calculus! (Don't worry, Calculus will materialize in due course.) Let's do a quic…
By Po-Shen Loh, Alex Federspiel
- Explanation #3326
By Expii Guest, Moor Xu
- The formula for the Chain Rule for single-variable functions is f(g(x))=f′(g(x))g′(x)f(g(x))=f'(g(x))g'(x)f(g(x))=f′(g(x))g′(x). If we have an equation f(g(x))=(x+2)3f(g(x))=(x+…
By Maya Schwartz, Alex Federspiel
- Suppose we need to take the derivative of 8x2+10\sqrt{8x^2+10}8x2+10. We can do this by using the chain rule. In order to complete the chain rule it is always easier to rewrite…
By Kyle Zigner
- Here is a worksheet containing chain rule problems. Answers are at the bottom. Good luck!
By Freya Edholm
- Derivative of Inverse If you got the answer, great job. Let's walk through the solution. The chain rule allows us to find the derivatives of compositions of functions. Indeed, l…
By Ananya Kumar, Alex Federspiel
- (Videos) Chain Rule Problems by PatrickJMT The Chain Rule is a very simple and important rule in derivatives that can get very complicated and messy very quickly. Below are a bu…
By Alex Federspiel
- (Videos) Using Chain, Product, and Quotient Rules to Find Derivatives by PatrickJMT Here's a video by patrickJMT showing you how to use the chain, product, and quotient rules to…
By Alex Federspiel
- (Video) Chain Rule for Finding Derivatives by PatrickJMT Here's a video by patrickJMT explaining the chain rule and how to use it to find derivatives. The chain rule can be thou…
By Alex Federspiel
- (Videos) Chain Rule Examples by PatrickJMT Here's a video by patrickJMT where he works through some examples involving using the chain rule to find derivatives. Recall: y=f[g(x)…
By Alex Federspiel
- (Videos) Harder Examples of the Chain Rule by PatrickJMT Hate yourself? Well here's a video by patrickJMT going over some harder chain rule examples. These can get long, messy, …
By Alex Federspiel
- (Videos) Using Chain, Product, and Quotient Rule to Find Derivatives by PatrickJMT Here's a video by patrickJMT showing you how to find the derivative of functions where you nee…
By Alex Federspiel
- (Video) Complicated Derivative Examples by PatrickJMT Here's a video by patrickJMT going over some derivative problems that have a mix of different rules. The first problem he l…
By Alex Federspiel
- (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
Problems 21
- What is the slope of f(x)=1000x2+1f(x) = \frac{1000}{x^2+1}f(x)=x2+11000 at x=3x = 3x=3?
- What is the derivative of y=(4x+1)2y=(4x+1)^2y=(4x+1)2?
- 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)…
- Suppose f(x)−f(2x)f(x) - f(2x)f(x)−f(2x) has derivative 555 at x=1x=1x=1 and 777 at x=2x=2x=2. Find the derivative of f(x)−f(4x)f(x) - f(4x)f(x)−f(4x) at x=1x=1x=1. Source: HMMT…
- What is the derivative of f(g(x))f(g(x))f(g(x))?
- Calculate f′(x)f'(x)f′(x) if f(x)=(2x+1)100f(x) = (2x+1)^{100}f(x)=(2x+1)100.
- Calculate f′(x)f'(x)f′(x) if f(x)=sin(cos(x))f(x)=\sin(\cos(x))f(x)=sin(cos(x)).
- 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).
- Calculate f′(x)f'(x)f′(x) if f(x)=arctan(sec(x))f(x)=\arctan(\sec(x))f(x)=arctan(sec(x)).
- Differentiate f(t)=13t2−5t+8f(t) = \sqrt{13t^2 - 5t + 8}f(t)=13t2−5t+8.
- Differentiate y=(1−3x+8x5)20y = (1-3x+8x^5)^{20}y=(1−3x+8x5)20.
- Differentiate eeexe^{e^{e^x}}eeex.
- What's the derivative of f(x)=axf(x) = a^xf(x)=ax?
- What is ddx 1000x2+1\frac{d}{dx}\; \frac{1000}{x^2+1}dxdx2+11000
- Let aaa and bbb be real numbers. What is the derivative of y=(lnxa)by=(\ln x^{a})^{b}y=(lnxa)b
- Find the derivative of y=ln(sin2x3)y=\ln(\sin2x^{3})y=ln(sin2x3)
- What is f′(x)f'(x)f′(x), where f(x)=ln(ln(x))f(x) = \ln(\ln(x))f(x)=ln(ln(x))
- What are the critical points for the function f(x)=ln(cosx)f(x)=\ln(\cos x)f(x)=ln(cosx) on the interval π2<x<3π2\frac{\pi}2<x<\frac{3\pi}22π<x<23π
- What are the critical points for the function f(x)=e2x+5f(x)=e^{2x+5}f(x)=e2x+5
- Consider this function which models an arc with endpoints at (±1,0):(\pm 1, 0):(±1,0): y=231−x2.y = \frac{2}{3}\sqrt{1 - x^2}.y=321−x2. Convert this function to polar coordina…