Explanation Given: The equation of the curve is y 5 = 32 x 4 . Also, the point is ( 1 , 2 ) . Formula used: Chain Rule: If f and g are functions and y = [ g ( x ) ] , then y ′ = f ′ [ g ( x ) ] ⋅ g ′ ( x ) provided that f ′ [ g ( x ) ] and g ′ ( x ) exists. And, The equation of tangent line is given as: y − y 1 = m ( x − x 1 ) Here, m is the slope, x 1 and y 1 are the given points. The equation for slope is: m = d y d x . Calculation: Implicit Differentiation: This differentiation is used to evaluate d y d x , when y is defined implicitly as a function of x . The differentiation is carried out in two steps: Step 1: Both the sides of the given equation is differentiated with respect to x . Since, y is the function of x , so always use the chain rule. d d x ( y 5 ) = d d x ( 32 x 4 ) 5 y 4 d y d x = 32 d d x ( x 4 ) 5 y 4 d y d x = 32 d d x ( 4 x 3 ) 5 y 4 d y d x = 128 x 3 Step 2: Solve the resultant equation for d y d x

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Chapter 12.4, Problem 32E

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To calculate: The equation of the tangent line at the given point for the given curve, y5=32x4.

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Chapter 12.4, Problem 31E

Chapter 12.4, Problem 33E

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The equation of the tangent line at the given point for the given curve, y 5 = 32 x 4 .

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