yr2 chap 9 differentiation

• sin x differentiates to cos x
• cos x differentiates to -sin x
• you need to be able to differentiate sin x and cos x using first principles, to do this use the formula for first principles and whichever function as f(x), then use the double angle formulae to expand the (x+h), then use small angle approximations of sin x and cos x to work out what the expression is equal to as h tends to zero
• ln x differentiates to 1/x
• e^kx differentiates to k(e^kx)
• a^kx differentiates to a^kx (k lna)
• you need to be able to prove that a^kx differentiates to a^kx (k lna), to do this take ln of a^kx and then raise e to the power of ln a^kx, because the e and the ln cancel each other out, then use the power rule to write e^(lna^kx) as e^(kx lna) then use the fact that e^kx differentiates to k(e^kx) to differentiate e^(kx lna) to get (k lna)e^(kx lna), use the power rule to rewrite this as (k lna)e^(lna^kx), then use the fact that e and ln cancel out to get a^kx (k lna)
• when a problem involves a function of a function, use the chain rule
• the chain rule is: dy/dx = dy/du * du/dx
• to use the chain rule, write y = f(u) where u = ax, find dy/du, find du/dx, then use the fact that dy/dx = dy/du * du/dx to get dy/dx
• when a problem involves the product of two functions, use the product rule
• the product rule is: where y = uv, dy/dx = u dv/dx + v du/dx
• to use the product rule, write out u and v in terms of x, then find du/dx and dv/dx, then use the fact that dy/dx = u dv/dx + v du/dx to get dy/dx
• when a problem involves a fraction of two functions, use the quotient rule
• the quotient rule is: dy/dx = (v du/dx - u dv/dx) / v^2
• to use the quotient rule, write out u and v in terms of x, then find du/dx and dv/dx, then use the fact that dy/dx = (v du/dx - u dv/dx) / v^2 to get dy/dx
• if asked to prove any of the trigonometric function differentiations from the formula book, rewrite the function in terms of sin x and cos x and then use the product rule or quotient rule to differentiate it
• the rule for parametric differentiation is that dy/dx = dy/dt / dx/dt
• in implicit differentiation, to differentiate something with y in, imagine the y is an x and write what that would be then add dy/dx
• in implicit differentiation, to differentiate something more complicated with y in, use the product rule or quotient rule
• to work out whether a curve is concave or convex on a given interval, use the second derivative
• if the second derivative is less than or equal to zero for all values in an interval, the curve is concave for that interval

  (remember as caves are underground)
• if the second derivative is greater than or equal to zero for all values in an interval, the curve is convex for that interval

  (remember as caves are underground)
• a point of inflection is a point where the second derivative changes sign
• to find a point of inflection, show that the second derivative equals zero at that point and show that the second derivative is positive on one side and negative on the other side of that point
• to form a differential equation, write down everything the question tells you such as what is proportional to what, then rearrange this into a form with two integrals equal to each other by matching the variables together
• when making differential equations, remember to include the constant of proportionality k if you removed the proportional sign