Power Rule for Derivatives: Integer Exponents
Read this topicExplanations 3
- How can we find the derivative of axa^xax (with respect to a)? First we take a look at the definition of a derivative: It is the limit as hhh approaches 000 of f(a+h)−f(a)h\frac…
By Vincent Huang, Joshua Siktar
- How can we find the derivative of f(x)=xnf(x)=x^nf(x)=xn , where n is an integer? We'll need to take a look at the definition of a derivative- It is the limit as h approaches 0 …
By Vincent Huang
- Power rule for integer exponents proved in 2 methods One of the most basic differentiation formulas is ddxxn=nxn−1.\frac{d}{dx} x^n = n x^{n-1}.dxdxn=nxn−1. (For example, ddxx1…
By Misha Lavrov, Frank Li, Victor Wang, Moor Xu
Problems 14
- What is the derivative ddx3x2\frac{d}{dx}3x^{2}dxd3x2?
- What is the second derivative of f(x)=4x7f(x)=4x^7f(x)=4x7?
- 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…
- Compute the 5th derivative of f(x)=x5f(x) = x^5f(x)=x5.
- The function f(x)=x3f(x) = x^3f(x)=x3 has a critical point at x=0x=0x=0. Classify this critical point into one of the following categories:
- Find the instantaneous rate of change for f(x)=x3f(x)=x^3f(x)=x3 at x=6x=6x=6.
- If f(x)=x3f(x)=x^3f(x)=x3 find f′(2)f'(2)f′(2).
- If f(x)=6x2f(x)=6x^2f(x)=6x2 find f′(5)f'(5)f′(5).
- If f(x)=x2f(x)=x^2f(x)=x2 find f′′′(53)f'''(53)f′′′(53).
- If f(x)=x3f(x)=x^3f(x)=x3 find f′′′(231)f'''(231)f′′′(231).
- True or False\text{True or False}True or False === If f(x)=xnf(x)=x^nf(x)=xn where n∈Nn\in\mathbb{N}n∈N, then f(n)=n!f^{(n)}=n!f(n)=n!
- Find the intervals on which the function f(x)=x2lnx2f(x)=x^2\ln x^2f(x)=x2lnx2 is increasing and decreasing.
- 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.
- Select all the functions that only have x=0x=0x=0 as their critical point.