Differentiate \( y=x\sin x \) with respect to \( x \).
\( \dfrac{dy}{dx}=\sin x + x\cos x \).
Further Differentiation
10 exam-style questions on product/quotient rules, trig/exp/ln, implicit and parametric differentiation. Tap Show answer for full working.
3 Easy4 Medium3 Hard
Find \( \dfrac{dy}{dx} \) when \( y=\dfrac{x}{1+x} \).
\( \dfrac{(1+x)-x}{(1+x)^2}=\dfrac{1}{(1+x)^2} \).
Differentiate \( y=e^{3x} \) and \( y=\ln(2x) \).
\( 3e^{3x} \) and \( \dfrac{1}{x} \).
Find the gradient of the curve \( y=x^2\ln x \) at the point where \( x=e \).
\( \dfrac{dy}{dx}=2x\ln x + x \). At \( x=e \): \( 2e\cdot1 + e=3e \).
Differentiate \( y=\cos(x^2) \) with respect to \( x \).
\( -2x\sin(x^2) \).
Given \( x^2 + y^2 = 25 \), find \( \dfrac{dy}{dx} \) in terms of \( x \) and \( y \).
\( 2x + 2y\dfrac{dy}{dx}=0 \) ⇒ \( \dfrac{dy}{dx}=-\dfrac{x}{y} \).
Find the equation of the tangent to \( y=\sin x \) at \( x=\dfrac{\pi}{3} \).
Gradient \( \cos(\pi/3)=1/2 \). Point \( (\pi/3, \sqrt{3}/2) \). \( y-\sqrt{3}/2=\dfrac12(x-\pi/3) \).
Find \( \dfrac{d^2y}{dx^2} \) when \( y=x e^{2x} \).
\( \dfrac{dy}{dx}=e^{2x}+2xe^{2x}=e^{2x}(1+2x) \). \( \dfrac{d^2y}{dx^2}=2e^{2x}(1+2x)+2e^{2x}=e^{2x}(4x+4) \).
The curve \( xy + y^2 = 6 \) passes through \( (1,2) \). Find the equation of the normal at this point.
Diff: \( y + x y' + 2y y'=0 \) ⇒ \( y'=-\dfrac{y}{x+2y} \). At \( (1,2) \): \( y'=-\dfrac{2}{1+4}=-\dfrac{2}{5} \). Normal gradient \( 5/2 \). \( y-2=\dfrac{5}{2}(x-1) \) ⇒ \( 2y-4=5x-5 \) ⇒ \( 5x-2y-1=0 \).
Find the stationary points of \( y=x^2 e^{-x} \) and determine their nature.
\( \dfrac{dy}{dx}=e^{-x}(2x-x^2)=xe^{-x}(2-x) \). Critical: \( x=0 \) or \( x=2 \). \( y(0)=0 \), \( y(2)=4e^{-2} \). Second deriv or sign: local min at \( (0,0) \), local max at \( (2,4/e^2) \).