Differentiation

• Differentiation is used to calculate the gradient of a function at different points.
• If y = x ^n, dy/dx = n x ^(n - 1).
• If y = a x ^ n, dy/dx = a n x ^ (n - 1).
• The derivative of a constant is zero.
• A function is increasing is the derivative is positive.
• A function is decreasing if the derivative is negative.
• A stationary point occurs when the derivative is zero.
• A local maximum occurs when the second derivative is negative.
• A local minimum occurs when the second derivative is positive.
• When the second derivative is zero, the point could be a maximum, minimum or point of inflection.
• When y = sin a x, dy/dx = a cos a x.
• When y = cos a x, dy/dx = -a sin a x.
• When y = e ^ a x, dy/dx = a e ^ a x.
• When y = a ^ k x, dy/dx = a ^(k x) (k ln a).
• When y = ln x, dy/dx = 1/x.
• dy/dx = dy/du x du/dx = 1/(dx/dy).
• If y = uv, dy/dx = u dv/dx + v du/dx.
• If y = u/v, dy/dx = (v du/dx - u dv/dx)/v ^2.
• If y = tan a x, dy/dx = a sec^2 a x.
• If y = cosec a x, dy/dx = -a cosec a x cot a x.
• If y = cot a x, dy/dx = -a cosec ^2 a x.
• If y = sec a x, dy/dx = a sec a x tan a x.
• d/dx f(y) = f' (y) dy/dx.
• f(x) is concave if the second derivative is negative.
• f(x) is convex if the second derivative is positive.
• A point of inflection is where the second derivative changes sign.