Power Rule for Derivatives: Rational Exponents
Read this topicExplanations 3
- Now we know that ddxxn=nxn−1\frac{d}{dx} x^n = n x^{n-1}dxdxn=nxn−1 when nnn is a positive integer. Is the same thing true for other numbers? With some work, we can extend this…
By Misha Lavrov
- How can we find the derivative of xpqx^{\frac{p}{q}}xqp, where p, q are positive integers? We let p=qrp=qrp=qr to keep things simple. Intuitively, the answer should be rxr−1rx^…
By Vincent Huang, Moor Xu
- For f(x)=xpqf(x)=x^\frac pqf(x)=xqp, we can use logarithmic differentiation to find ddx(f(x))\frac d{dx}(f(x))dxd(f(x)): ddx(ln(f(x)))=ddx(pqln(x))\frac d{dx}(\ln(f(x))) = \…
By Alexander Duan
Problems 11
- What is the derivative of 6x5/36 x^{5/3}6x5/3?
- Differentiate 2x+3x\sqrt 2 x + \sqrt{3x}2x+3x.
- If h(x)=2x+1h(x) = \sqrt{2x+1}h(x)=2x+1, find h′(4)h'(4)h′(4).
- Find the linear approximation of f(x)=x56f(x)=x^{\frac56}f(x)=x65 at a=64a=64a=64.
- Use differentials to estimate 170\sqrt{170}170.
- Use differentials to estimate 9239^{\frac23}932.
- Harry, on his way to Hogwarts, arrived at platform 9349^{\frac34}943. While waiting for the train, he became curious as to what the value of 9349^{\frac34}943 actually was. Sa…
- Use differentials to estimate 13.74\frac{1}{\sqrt{3.74}}3.741.
- What is the differential equation for 4−x\sqrt{4-x}4−x?
- Use linearization to estimate 13.74\frac{1}{\sqrt{3.74}}3.741.
- Find the intervals on which the function f(x)=x2x+2f(x)=x^2\sqrt{x+2}f(x)=x2x+2 is increasing and decreasing.