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Differentiating f(x) using the Chain Rule, we have
f
(x)=−
1
2
(25 − x
2
)
−1/2
(25 − x
2
)
= −
1
2
−2x
√
25 − x
2
=
=
x
√
25 − x
2
,f
(3) =
3
4
.
and, as in Solution 1, the equation of the tangent is 3x − 4y =25.
Remark 3
Examples 1, 2 illustrate that even when it is possible to solve
an equation explicitly for y in terms of x, it may be easier to use
implicit differentiation.
Remark 4
The expression y
= −
x
y
gives the derivative in terms of both x
and y. It is correct no matter which function y is determined by the
given equation.
Exercise 1
• Find y
if x
3
+ y
3
=6xy.
• Find the tangent to this curve at the point (3, 3).
• At what points on the curve is the tangent line horizontal or
vertical?
Example 14
We can use the Chain Rule to differentiate an exponential function
with any base a>0. Recall that a = e
ln a
.Soa
x
=(e
ln a
)x = e
(ln a)x
and the Chain Rule gives
(a
x
)
=(e
(ln a)x
)
= e
(ln a)x
[(ln a)x]
= e
(ln a)x
ln a = a
x
ln a
because ln a is a constant. So we have the formula
d
dx
(a
x
)=a
x
ln a.
23
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