Derivative of Composition of Functions

Calculus Level 2

There is a certain function f f such that the tangent line at x = 1 2 x=\frac{1}{2} is y = 2 x + 1. y=2x+1.

Given that g ( x ) = f ( sin x ) , g(x)=f(\sin x), find the sum of all possible values of d g d x \frac{dg}{dx} when evaluated at x = π 6 x=\frac{\pi}{6} .

1 2 -\frac{1}{2} 0 1 2 \frac{1}{2} π 6 \frac{\pi}{6} 3 2 \frac{\sqrt{3}}{2} 3 \sqrt{3} 2 2 Undefined

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

C .
Aug 21, 2018

g ( π 6 ) = d g d x ( π 6 ) = [ d d x g ( x ) ] ( π 6 ) = [ d d x f ( s i n ( x ) ) ] ( π 6 ) = [ f ( s i n ( x ) ) × d d x s i n ( x ) ] ( π 6 ) = [ f ( s i n ( x ) ) × c o s ( x ) ] ( π 6 ) g'(\frac\pi6)=\frac{dg}{dx}(\frac\pi6)=[\frac{d}{dx}g(x)](\frac\pi6)=[\frac{d}{dx}f(sin(x))](\frac\pi6)=[f'(sin(x))\times\frac{d}{dx}sin(x)](\frac\pi6)=[f'(sin(x))\times cos(x)](\frac\pi6) = f ( s i n ( π 6 ) ) × c o s ( π 6 ) = f ( 1 2 ) × 3 2 = 2 × 3 2 = 3 =f'(sin(\frac\pi6))\times cos(\frac\pi6)=f'(\frac12)\times\frac{\sqrt3}2=2\times\frac{\sqrt3}2=\sqrt3

Note that f ( 1 2 ) = 2 f'(\frac12)=2 since the value of the derivative is the slope of the tangent, and the line equation given is in slope-intercept form, y ( x ) = m x + y ( 0 ) y(x)=mx+y(0) ... Alternatively, the tangent passes through the ( 0 , 1 ) (0,1) and the ( 1 2 , 2 ) (\frac12,2) points and has a slope of 2 1 1 2 0 = 2 \frac{2-1}{\frac12-0}=2 .

Nick Turtle
May 2, 2018

We have that g ( x ) = f ( sin x ) g(x)=f(\sin{x})

Differentiate both sides with respect to x x (use the chain rule on the right-hand side) to get d g d x = cos x d d ( sin x ) f ( sin x ) \frac{dg}{dx}=\cos{x} \frac{d}{d(\sin{x})}f(\sin{x})

Now, simply evaluate the above at x = π 6 x=\frac{\pi}{6} to get = 3 =\sqrt{3}

using the fact that the instantaneous slope of f ( x ) f(x) with respect to x x at x = 2 x=2 is 2 2 .

slope at x = 1 2 \frac12 you mean...

C . - 2 years, 9 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...