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Section 3.6 : Derivatives of Exponential and Logarithm Functions

7. Find the tangent line to $$f\left( x \right) = {7^x} + 4{{\bf{e}}^x}$$ at $$x = 0$$.

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We know that the derivative of the function will give us the slope of the tangent line so we’ll need the derivative of the function.

$f'\left( x \right) = {7^x}\ln \left( 7 \right) + 4{{\bf{e}}^x}$ Show Step 2

Now all we need to do is evaluate the function and the derivative at the point in question.

$f\left( 0 \right) = 5\hspace{0.25in}\hspace{0.25in}f'\left( 0 \right) = \ln \left( 7 \right) + 4 = 5.9459$ Show Step 3

Now all that we need to do is write down the equation of the tangent line.

$y = f\left( 0 \right) + f'\left( 0 \right)\left( {x - 0} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{5 + \left( {\ln \left( 7 \right) + 4} \right)x = 5 + 5.9459x}}$