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3.4 Differentiation

Syllabus
9709–2028–2029
Topic
3.4
Level
A2

Learning objectives

3.4.1Advanced differentiation• use the derivatives of ex, ln x, sin x, cos x, tan x, tan-1 x, together with constant multiples, sums, differences and composites Derivatives of sin-1 x and cos-1 x are not required.3.4.2Product and quotient rules• differentiate products and quotients e.g. x x 32 24 - +, x2 ln x, xe1 - x2.3.4.3Differentiation 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 A2 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)
tan⁡−1x\tan^{-1}x 1/(1+x2)1/(1+x^2)

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.Forinversetangent:For inverse tangent:\frac d{dx}\tan^{-1}(g)=\frac{g'}{1+g^2}.

\frac d{dx}\ln(1+x^2)=\frac{2x}{1+x^2},\qquad \frac d{dx}\tan^{-1}(3x)=\frac{3}{1+9x^2}.

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.

Do not confuse tan⁡−1x\tan^{-1}x with 1/tan⁡x1/\tan x. Derivatives of sin⁡−1x\sin^{-1}x and cos⁡−1x\cos^{-1}x are explicitly not required.

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.