STEP I 1994 question 2 solution

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Part i)

$\frac{\text{d}y}{\text{d}x} = ax^{(a-1)}$

Part ii) $\ y = a^x}$

$\ln y = \ln(a^x)$

$\ln y = x \ln a$

$\frac{\text{d}y}{\text{d}x} \frac{1}{y} = \ln a$

$\frac{\text{d}y}{\text{d}x} = y \ln a$

$\frac{\text{d}y}{\text{d}x} = a^x \ln a$

Part iii)

$\ln y = \ln (x^x)$

$\ln y = x \ln x$

$\frac{\text{d}y}{\text{d}x} \frac{1}{y} = \ln x + 1$

$\frac{\text{d}y}{\text{d}x} = x^x( \ln x + 1)$

Part iv) $y = x^{x^x}$

$\ln y = \ln (x^{x^x})$

$\ln y = x^x \ln x$

$\frac{\text{d}y}{\text{d}x} \frac{1}{y} = x^x (\ln x + 1) \lnx + \frac{x^x}{x}$ (From part iii)

$\frac{\text{d}y}{\text{d}x} \frac{1}{y} = x^x ((\ln x)^2 + \ln x) + x^{(x-1)}$

$\frac{\text{d}y}{\text{d}x} = yx^x(( \ln x)^2 + \ln x + \frac{1}{x})$

$\frac{\text{d}y}{\text{d}x} = x^{(x^x + x)}(( \ln x)^2 + \ln x + \frac{1}{x})$

Part v)

$y = x^{x^2}$

$\ln y = \ln (x^{x^2})$

$\ln y = x^2 \ln x$

$\frac{\text{d}y}{\text{d}x} \frac{1}{y} = 2x \ln x + x$

$\frac{\text{d}y}{\text{d}x} = x^{x^2} (2x \ln x + x)$

$\frac{\text{d}y}{\text{d}x} = x^{(x^2+1)} (2 \ln x + 1)$

Solution by zrancis

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Category: STEP Papers

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