Given the functions f ( x ) = tan ⁡ ( π x ) and g ( x ) = cos ⁡ ( π x 3 ), compute the derivative of (f*g)(x).

Set up the composite result function.f(g(x))Evaluate&nbsp;f(cos(πx3))&nbsp;by substituting in the value of&nbsp;g&nbsp;into&nbsp;f.f(cos(πx3))=tan(π(cos(πx3)))Rewrite&nbsp;tan(πcos(πx3))&nbsp;in terms of sines and cosines.f(cos(πx3))=sin(πcos(πx3))cos(πcos(πx3))Convert from&nbsp;sin(πcos(πx3))cos(πcos(πx3))&nbsp;to&nbsp;tan(πcos(πx3)).f(cos(πx3))=tan(πcos(πx3))

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College
Calculus and Analysis
Differential Calculus
Derivatives

arbixerwoxottdrp1l

Answered question

2022-04-10

Given the functions

f (x) = \tan (π x)

and

g (x) = \cos (\frac{π x}{3})

, compute the derivative of (f*g)(x).

Answer & Explanation

Finnegan Zimmerman

Beginner2022-04-11Added 16 answers

Set up the composite result function.

$f (g (x))$

Evaluate $f (\cos (\frac{π x}{3}))$ by substituting in the value of $g$ into $f$ .

$f (\cos (\frac{π x}{3})) = \tan (π (\cos (\frac{π x}{3})))$

Rewrite $\tan (π \cos (\frac{π x}{3}))$ in terms of sines and cosines.

$f (\cos (\frac{π x}{3})) = \frac{\sin (π \cos (\frac{π x}{3}))}{\cos (π \cos (\frac{π x}{3}))}$

Convert from $\frac{\sin (π \cos (\frac{π x}{3}))}{\cos (π \cos (\frac{π x}{3}))}$ to $\tan (π \cos (\frac{π x}{3}))$ .

$f (\cos (\frac{π x}{3})) = \tan (π \cos (\frac{π x}{3}))$

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