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The chain rule. The chain rule is a powerful method used to differentiate composite functions (i.e. expressions where one function is contained in another function). An example of such a function would be. 2−3 +1, where the function 2 − 3 + 1 is contained inside the function .
Derivatives Rules. Power Rule \frac {d} {dx}\left (x^a\right)=a\cdot x^ {a-1} Derivative of a constant \frac {d} {dx}\left (a\right)=0. Sum Difference Rule \left (f\pm g\right)^'=f^'\pm g^'.
2 x - 9 y + x 3 y. 2 = sin ( y ) + 11 x . Remember y = y ( x ) here, so products/quotients of x and y will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y¢ (from the chain rule).
2 x - 9 y + x 3 y. 2 = sin ( y ) + 11 x . Remember y = y ( x ) here, so products/quotients of x and y will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y¢ (from the chain rule).
Derivatives Moderate Chain Rule 1. dx d cos 2x 2. dx d 2x +5 3. dx d 3 4. dx d sin x 5. dx d e 6. dx d ln x −5x 7. dx d 2x −1 8. dx d 4x−3 9. dx d 2x+6 10. dx d tan 2x
Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the function. Many answers: Ex y = ( ( ( 2 x + 1)5 + 2)6 + 3)7 dy = 7 ( ( ( 2 x + 1)5 + 2)6 + 3)6 ⋅ 6 ( ( 2 x + 1)5 + 2)5 ⋅ 5 ( 2 x + 1)4 ⋅ 2 dx.
Hyperbolic Derivatives: (sinh( )) = cosh( ) (cosh( )) = sinh( ) Product Rule: ( ⋅ )′ = ′ ⋅ + ⋅ ′. ′⋅ − ′⋅. 2. ′. Quotient Rule: ( ) = ( ) Chain Rule: = ⋅.
