Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \text{If }f(x)=\sqrt{3x}\text{, then the index form is:} \\ f(x)=3x^{\frac{1}{2}}\text{?} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer:} \\ f(x)=\sqrt{3}{x}^{\frac{1}{2}}\text{ or }{3}^{\frac{1}{2}}{x}^{\frac{1}{2}} \\ \text{The square root applies to the }3\text{ as well. If differentiating, the coefficient is }\sqrt{3}\text{, not }3\text{.} \end{gathered}$$
• $\text{To find the coordinates of the point where the gradient of the curve is zero, what algebraic step must you take?}$
  $$\begin{gathered} \text{1. Find the derivative }{f}^{\prime }(x)\text{ or }\frac{dy}{dx}\text{.} \\ \text{2. Set }{f}^{\prime }(x)=0\text{.} \\ \text{3. Solve the resulting equation for }x\text{.} \\ \text{4. Substitute }x\text{ back into the original function }f(x)\text{ to find }y\text{.} \end{gathered}$$
• $\text{What is the formula for the derivative from first principles (}{f}^{\prime }(x)\text{)?}$
  $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$
• $$\begin{gathered} \text{You are given }y=5x^3\text{ and asked to find the rate of change of the gradient.} \\ \text{What mathematical operation do you perform?} \end{gathered}$$
  $$\begin{gathered} \text{Find the Second Derivative (}\frac{d^{2}y}{d{x}^2}\text{).} \\ \text{1. Differentiate to get gradient: }\frac{dy}{dx}=15x^2 \\ \text{2. Differentiate again: }\frac{d^{2}y}{d{x}^2}=30x \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{To differentiate }y=(2x-3{)}^2\text{, the result is:} \\ \frac{dy}{dx}=2(2x-3) \end{gathered}$$
  $$\begin{gathered} \text{❌ False (in the context of P1 methods)} \\ \text{✅ Correct method:} \\ \text{Expand the brackets first:} \\ y=4x^2-12x+9 \\ \text{Then differentiate term by term:} \\ \frac{dy}{dx}=8x-12 \end{gathered}$$
• $$\begin{gathered} \text{What is the first step to differentiate the following expression?} \\ y=\frac{x^3+4x^2-3}{\sqrt{x}}\text{?} \end{gathered}$$
  $$\begin{gathered} \text{Split the fraction (divide each term by the denominator) and write in index form }a{x}^n\text{.} \\ y=\frac{x^3}{x^{0.5}}+\frac{4x^2}{x^{0.5}}-\frac{3}{x^{0.5}} \\ y=x^{2.5}+4x^{1.5}-3x^{-0.5} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{When differentiating }y=5 \\ {\pi }^2\text{, the derivative is:} \\ \frac{dy}{dx}=10 \\ \pi \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }0 \\ 5 \\ {\pi }^2\text{ is a constant (it contains no }x\text{ terms). The derivative of any constant is zero.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If }y=\frac{1}{2x^3}\text{, then:} \\ \frac{dy}{dx}=-3\times 2x^{-4}=-6x^{-4} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct method:} \\ \text{Rewrite first: }y=\frac{1}{2}{x}^{-3}\text{ (The coefficient }\frac{1}{2}\text{ is a constant multiplier).} \\ \text{Then differentiate:} \\ \frac{dy}{dx}=\frac{1}{2}\times (-3)x^{-4}=-\frac{3}{2}{x}^{-4} \end{gathered}$$