Explained: "Find the derivatives of the functions defined as..."

PEEWSRIGWETRYqx

Answered question

2021-12-06

Find the derivatives of the functions defined as follows.

y = - 4 \cos (7 x^{2} - 4)

Answer & Explanation

lenkiklisg7

Beginner2021-12-07Added 29 answers

Step 1
The given function is,

y = - 4 \cos (7 x^{2} - 4)

If a function is the function of some other function as y=f(g(x)), then we use the chain rule of differentiation to find the derivative of the function.

\frac{d y}{d x} = \frac{d}{d (g (x))} [f (g (x))] \cdot \frac{d}{d x} [(g (x))]

Step 2
Hence, using chain rule of differentiation, we get the derivative of the given function as,

\frac{d y}{d x} = \frac{d}{d x} (- 4 \cos (7 x^{2} - 4))

= [\frac{d}{d (7 x^{2} - 4)} [- 4 \cos (7 x^{2} - 4)]] \cdot [\frac{d}{d x} [7 x^{2} - 4]]

= - 4 [\frac{d}{(7 x^{2} - 4)} [\cos (7 x^{2} - 4)]] \cdot [7 (2 x - 0)]

= - 4 (- \sin (7 x^{2} - 4)) (14 x)

= 56 x \sin (7 x^{2} - 4)

Therefore, the derivative of the given function is

56 x \sin (7 x^{2} - 4)

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