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2.4 Differentiation

Syllabus
9709–2028–2029
Topic
2.4
Level
AS

Learning objectives

2.4.1Advanced differentiation• use the derivatives of ex, ln x, sin x, cos x, tan x, together with constant multiples, sums, differences and composites2.4.2Product and quotient rules• differentiate products and quotients e.g. x x 32 24 - +, x2 ln x, xel - x2.2.4.3Differentiation rules• find and use the first derivative of a function which is defined parametrically or implicitly. e.g. x = t - e2t, y = t + e2t. e.g. x2 + y2 = xy + 7. Including use in problems involving tangents and normals.

Attach the inner derivative to each P2 base derivative

f(x)f(x) f′(x)f'(x)
exe^x exe^x
ln⁡x\ln x 1/x1/x (x>0x>0)
sin⁡x\sin x cos⁡x\cos x
cos⁡x\cos x −sin⁡x-\sin x
anxan x sec⁡2x\sec^2x (where defined)

For differentiable $g$: rac d{dx}e^{g}=g'e^g,\quad rac d{dx}\ln g= rac{g'}g,\quad rac d{dx}\sin g=g'\cos g, rac d{dx}\cos g=-g'\sin g,\quad rac d{dx} an g=g'\sec^2g.

rac d{dx}\ln(1+x^2)= rac{2x}{1+x^2},\qquad rac d{dx}[3e^{2x}-\cos(4x)]=6e^{2x}+4\sin(4x).

Identify the outer function, write its base derivative with the inner expression unchanged, multiply by the inner derivative, then combine constant multiples, sums and differences.

The derivative of ln⁡g\ln g is g′/gg'/g, not 1/ln⁡g1/\ln g. Products and quotients are handled in the next objective rather than smuggled into a sum/chain shortcut.

Differentiate each changing factor in a product or quotient

If $y=u(x)v(x)$,y'=u'v+uv'.If $y=u(x)/v(x)$ and $v e0$,y'= rac{u'v-uv'}{v^2}.

Label uu and vv, compute u′u' and v′v' separately using chain rules where needed, substitute without changing the quotient numerator order, then factor or simplify.

rac d{dx}(x^2e^x)=2xe^x+x^2e^x=e^x(x^2+2x).

rac d{dx}\left( rac{\sin x}{x}
ight)= rac{x\cos x-\sin x}{x^2},\qquad x
e0.

The product derivative is not u′v′u'v', and the quotient derivative is not u′/v′u'/v'. Logarithmic differentiation is not required by this objective.

Extract gradients from parametric and implicit definitions

Definition Route to dy/dxdy/dx
x=x(t), y=y(t)x=x(t),\ y=y(t) dy/dx=(dy/dt)/(dx/dt)dy/dx=(dy/dt)/(dx/dt) when $dx/dt
e0$
F(x,y)=0F(x,y)=0 differentiate both sides in xx, attach dy/dxdy/dx to every yy derivative, then collect

If $x=t-e^{2t}$ and $y=t+e^{2t}$, rac{dy}{dx}= rac{1+2e^{2t}}{1-2e^{2t}}wherever $dx/dt e0$.

For $x^2+y^2=xy+7$:2x+2y rac{dy}{dx}=y+x rac{dy}{dx},soso rac{dy}{dx}= rac{y-2x}{2y-x}when $2y-x e0$.

Find the parameter or point coordinates first, evaluate the tangent gradient mm, then use y−y0=m(x−x0)y-y_0=m(x-x_0). A non-vertical normal has gradient −1/m-1/m; handle horizontal/vertical tangent cases geometrically.

Parametric gradient is dy/dtdy/dt divided by dx/dtdx/dt, not the reverse. In implicit differentiation, d(y2)/dx=2y dy/dxd(y^2)/dx=2y\,dy/dx, not 2y2y.