Moderate -0.3 This is a straightforward differentiation question requiring the chain rule for the square root term and quotient rule for the rational term, followed by substitution of x=4. It's slightly easier than average because it's a direct application of standard techniques with no problem-solving element, though the combination of two rules and exact value calculation keeps it from being trivial.
Differentiate first term to obtain form \(k(4x-7)^{-\frac{1}{2}}\)
*M1
Any non-zero constant \(k\); M0 if this differentiation is carried out in the midst of some incorrect involved expression
Obtain \(2(4x-7)^{-\frac{1}{2}}\)
A1
Or (unsimplified) equiv
Attempt use of quotient rule or, after adjustment, product rule
*M1
For QR, allow numerator wrong way round but needs \(-\) sign in numerator; condone a single error such as absence of square in denominator, absence of brackets, …; for PR, condone no use of chain rule M0 if this differentiation is carried out in the midst of some incorrect involved expression
Obtain \(\frac{4(2x+1)-8x}{(2x+1)^2}\) or \(4(2x+1)^{-1}-8x(2x+1)^{-2}\)
A1
Or (unsimplified) equivs; give A0 if brackets absent unless subsequent calculation indicates their 'presence'
Substitute 4 into expression for first derivative so that (initially at least) exactness is retained
M1
Dep *M *M
Obtain \(\frac{58}{81}\)
A1
Answer must be exact. Note: using \(y=\sqrt{4x-7}+\frac{4}{2x+1}\): do not apply MR
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate first term to obtain form $k(4x-7)^{-\frac{1}{2}}$ | *M1 | Any non-zero constant $k$; M0 if this differentiation is carried out in the midst of some incorrect involved expression |
| Obtain $2(4x-7)^{-\frac{1}{2}}$ | A1 | Or (unsimplified) equiv |
| Attempt use of quotient rule or, after adjustment, product rule | *M1 | For QR, allow numerator wrong way round but needs $-$ sign in numerator; condone a single error such as absence of square in denominator, absence of brackets, …; for PR, condone no use of chain rule M0 if this differentiation is carried out in the midst of some incorrect involved expression |
| Obtain $\frac{4(2x+1)-8x}{(2x+1)^2}$ or $4(2x+1)^{-1}-8x(2x+1)^{-2}$ | A1 | Or (unsimplified) equivs; give A0 if brackets absent unless subsequent calculation indicates their 'presence' |
| Substitute 4 into expression for first derivative so that (initially at least) exactness is retained | M1 | Dep *M *M |
| Obtain $\frac{58}{81}$ | A1 | Answer must be exact. Note: using $y=\sqrt{4x-7}+\frac{4}{2x+1}$: do not apply MR |
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4 Find the exact value of the gradient of the curve
$$y = \sqrt { 4 x - 7 } + \frac { 4 x } { 2 x + 1 }$$
at the point for which $x = 4$.
\hfill \mbox{\textit{OCR C3 2013 Q4 [6]}}