10 Differentiation

10.3 Powers of x\bm x

Find dydx\Large\frac{dy}{dx} for each of the following:
  1. y=x2y=x^2
  2. y=1x3y=\displaystyle\frac{1}{x^3}
  3. y=17y=17
  4. y=7xy=7\sqrt{x}
Find f′(x)f'(x) for each of the following:
  1. f(x)=7x6f(x)=7x^6
  2. f(x)=5x−4f(x)=5x^{-4}
  3. f(x)=3x3f(x)=\displaystyle\frac{3}{\sqrt{x^3}}
  4. f(x)=7x25f(x)=7\sqrt[5]{x^2}
Evaluate the derivative of each curve at the specified location to determine the gradient of the curve at that point.
  1. y=x48y=\displaystyle\frac{x^4}{8} at x=3x=3
  2. f(x)=x−2f(x)=x^{-2} when x=−2x=-2
  3. y=xy=\sqrt{x} at the point (4,2)(4, 2)
  4. f(x)=4x1/3f(x)=\displaystyle\frac{4}{x^{1/3}} at x=−8x=-8