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3 years ago

Derivative for tan(x)

Chemistry

1 answer:

Maksim231197 [3]3 years ago

4 0

Answer:

sec² (x)

Explanation:

Use the trigonometric ratio definition of the tangent function and the quotient rule.

tan(x) = sin(x) / cos(x)

Quotient rule: the derivative of a quotient is:

[the denominator × the derivative of the numerator less the numerator × the derivative of the denominator] / [denominator]²

(f/g)' = [ g×f' - f×g'] / g²

So,

tan(x)' = [ sin(x) / cos(x)]'

[ sin(x) / cos(x)]' = [ cos(x) sin(x)' - sin(x) cos(x)' ] / [cos(x)]²

= [ cos(x)cos(x) + sin(x) sin(x) ] / [ cos(x)]²

= [ cos²(x) + sin²(x) ] / cos²(x)

= 1 / cos² (x)

= sec² (x)

The result is that the derivative of tan(x) is sec² (x)

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As long as the equation in question can be expressed as the sum of the three equations with known enthalpy change, its $\Delta H$ can be determined with the Hess's Law. The key is to find the appropriate coefficient for each of the given equations.

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