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Use implicit differentiation to find the slope of the tangent line to the curve defined by...

Question:

Use implicit differentiation to find the slope of the tangent line to the curve defined by {eq}4xy^6+9xy=26 {/eq} at the point {eq}(2,1) {/eq}.

Implicit Differentiation:

Using the implicit differentiation the problem will be solved where we will differentiate the equation using the product rule now let us consider we have the product of two functions f and g and then to differentiate it we will use the product rule.

Answer and Explanation:

To solve the problem we will use the implicit differentiation:

{eq}4xy^{6}+9xy=26 {/eq}

Now differentiating the equation:

{eq}4y^{6}+24xy^{5}\frac{\mathrm{d} y}{\mathrm{d} x}+9y+9x\frac{\mathrm{d} y}{\mathrm{d} x}=0 {/eq}

Now after rearranging we get the derivative:

{eq}\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{-9y-4y^{6}}{9x+24xy^{5}} {/eq}

Now let us plug-in the values we get:

{eq}\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{-13}{66} {/eq}


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