find the derivative of sin((3pi x)/25)cos((pi x)/25)+15

Answered question

2022-05-11

find the derivative of sin((3pi x)/25)cos((pi x)/25)+15

Answer & Explanation

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Skilled2022-05-13Added 403 answers

sin(3πx25)cos(πx25)+15

By the Sum Rule, the derivative of sin(3πx25)cos(πx25)+15 with respect to x is ddx[sin(3πx25)cos(πx25)]+ddx[15].

ddx[sin(3πx25)cos(πx25)]+ddx[15]

Evaluate ddx[sin(3πx25)cos(πx25)].

Differentiate using the Product Rule which states that ddx[f(x)g(x)] is f(x)ddx[g(x)]+g(x)ddx[f(x)] where f(x)=sin(3πx25) and g(x)=cos(πx25).

sin(3πx25)ddx[cos(πx25)]+cos(πx25)ddx[sin(3πx25)]+ddx[15]

Differentiate using the chain rule, which states that ddx[f(g(x))] is f(g(x))g(x) where f(x)=cos(x) and g(x)=πx25.

sin(3πx25)(−sin(πx25)ddx[πx25])+cos(πx25)ddx[sin(3πx25)]+ddx[15]sin(3πx25)(-sin(πx25)ddx[πx25])+cos(πx25)ddx[sin(3πx25)]+ddx[15]

Since π25π25 is constant with respect to xx, the derivative of πx25πx25 with respect to xx is π25ddx[x]π25ddx[x].

sin(3πx25)(-sin(πx25)(π25ddx[x]))+cos(πx25)ddx[sin(3πx25)]+ddx[15]

Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.

sin(3πx25)(-sin(πx25)(π251))+cos(πx25)ddx[sin(3πx25)]+ddx[15]

Differentiate using the chain rule, which states that ddx[f(g(x))] is f(g(x))g(x) where f(x)=sin(x) and g(x)=3πx25.

sin(3πx25)(-sin(πx25)(π251))+cos(πx25)(cos(3πx25)ddx[3πx25])+ddx[15]

Since 3π25 is constant with respect to x, the derivative of 3πx25 with respect to x is 3π25ddx[x].

sin(3πx25)(-sin(πx25)(π251))+cos(πx25)(cos(3πx25)(3π25ddx[x]))+ddx[15]

Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.

sin(3πx25)(-sin(πx25)(π251))+cos(πx25)(cos(3πx25)(3π251))+ddx[15]

Multiply π25 by 1.

sin(3πx25)(-sin(πx25)π25)+cos(πx25)(cos(3πx25)(3π251))+ddx[15]

Combine π25 and sin(πx25).

sin(3πx25)(-πsin(πx25)25)+cos(πx25)(cos(3πx25)(3π251))+ddx[15]

Combine sin(3πx25) and πsin(πx25)25.

-sin(3πx25)(πsin(πx25))25+cos(πx25)(cos(3πx25)(3π251))+ddx[15]

Multiply 3π25 by 1.

-sin(3πx25)πsin(πx25)25+cos(πx25)(cos(3πx25)3π25)+ddx[15]

Combine cos(3πx25) and 3π25.

-sin(3πx25)πsin(πx25)25+cos(πx25)cos(3πx25)(3π)25+ddx[15]

Move 3 to the left of cos(3πx25).

-sin(3πx25)πsin(πx25)25+cos(πx25)3cos(3πx25)π25+ddx[15]

Combine cos(πx25) and 3cos(3πx25)π25.

-sin(3πx25)πsin(πx25)25+cos(πx25)(3cos(3πx25)π)25+ddx[15]

Move 3 to the left of cos(πx25).

-sin(3πx25)πsin(πx25)25+3cos(πx25)cos(3πx25)π25+ddx[15]

Since 15 is constant with respect to x, the derivative of 15 with respect to x is 0.

-sin(3πx25)πsin(πx25)25+3cos(πx25)cos(3πx25)π25+0

Simplify.

-sin(3πx25)πsin(πx25)25+3cos(πx25)cos(3πx25)π25 and 0.

-sin(3πx25)πsin(πx25)25+3cos(πx25)cos(3πx25)π25

Reorder terms.

3cos(πx25)cos(3πx25)π25-sin(3πx25)πsin(πx25)25

Reorder factors in 3cos(πx25)cos(3πx25)π25-sin(3πx25)πsin(πx25)25.

 

3πcos(πx25)cos(3πx25)25-πsin(3πx25)sin(πx25)25 - Answer
 

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