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Worked solution

A nested derivative, correct to five significant figures

Evaluate ddx 23x2+sin⁡2(ex2)∣x=1\left.\dfrac{d}{dx}\,2^{3x^2+\sin^2(e^{x^2})}\right|_{x=1}, correct to 5 significant figures.

CJC · Discussion · Question 1

Answer

12.01612.016 (5 s.f.), using the analytically checked derivative. The calculator value printed in the key has a rounding discrepancy.

Analytical method

Write u(x)=3x2+sin⁡2(ex2)u(x)=3x^2+\sin^2(e^{x^2}). Apply the chain rule at every layer:

ddx2u(x)=2u(x)ln⁡2[6x+2sin⁡(ex2)cos⁡(ex2) ex2 2x]\dfrac{d}{dx}2^{u(x)}=2^{u(x)}\ln2[6x+2\sin(e^{x^2})\cos(e^{x^2})\,e^{x^2}\,2x].

At x=1x=1 this is 23+sin⁡2eln⁡2 [6+4esin⁡ecos⁡e]=12.0161368258…2^{3+\sin^2e}\ln2\,[6+4e\sin e\cos e]=12.0161368258\ldots.

The answer is 12.016 (5 s.f.).

How did this part go?

Calculator method and source check

The printed calculator method reports 12.01710163…12.01710163\ldots, rounded to 12.017, while the printed analytical method reports 12.01613683…12.01613683\ldots, rounded to 12.016. They disagree in the requested precision.

The analytical result above is independently confirmed. A central-difference estimate with step 0.001 reproduces the printed calculator value; reducing the step makes the estimate approach 12.0161368. Use the analytically checked value for the requested answer.

The source notes specify radian mode and entering [sin⁡(ex2)]2[\sin(e^{x^2})]^2 rather than treating the exponent as part of the sine input.

How did this part go?

Common mistakes

Use radians. Numerical differentiation can be too coarse for a five-significant-figure answer.

Solution notes

The printed calculator and analytical answers differ. The retained transcription preserves both; the worked answer uses the independently checked analytical value.

Read the hint