# How do you find the slope of the tangent line #y=(5x^2+7)^2# at x=1?

The slope of the tangent to the function y at x = 1 is the derivative of the function evaluated at x = 1

change u back into terms of x.

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To find the slope of the tangent line at a point on a curve, we can use the derivative of the function evaluated at that point.

The derivative of the given function ( y = (5x^2 + 7)^2 ) can be found using the chain rule, which states that if ( y = f(g(x)) ), then ( y' = f'(g(x)) \cdot g'(x) ).

Taking the derivative of ( f(g(x)) = g(x)^2 ) with respect to ( g(x) ), we get ( f'(g(x)) = 2g(x) ).

Taking the derivative of ( g(x) = 5x^2 + 7 ) with respect to ( x ), we get ( g'(x) = 10x ).

Now, we can find the derivative of the function ( y = (5x^2 + 7)^2 ) by applying the chain rule:

( \frac{dy}{dx} = 2(5x^2 + 7) \cdot 10x = 20x(5x^2 + 7) ).

To find the slope of the tangent line at ( x = 1 ), we evaluate the derivative at ( x = 1 ):

( \frac{dy}{dx} \Bigg|_{x=1} = 20(1)(5(1)^2 + 7) = 20(1)(5+7) = 20(1)(12) = 240 ).

Therefore, the slope of the tangent line at ( x = 1 ) is ( 240 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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