If y = \cos(3x) + e^x, what is \frac{dy}{dx}? — JAMB UTME

If $y = \cos(3x) + e^x$ , what is $\frac{dy}{dx}$ ?

-3\sin(3x) + e^x

Differentiating

y = \cos(3x) + e^x

term-by-term yields

\frac{dy}{dx} = -3\sin(3x) + e^x

. By the chain rule,

\frac{d}{dx}\cos(3x) = -3\sin(3x)

, while the derivative of

e^x

is simply

e^x

Differentiate the trigonometric term

\cos(3x)

using the chain rule

\frac{d}{dx}(\cos(3x)) = -\sin(3x) \cdot \frac{d}{dx}(3x) = -3\sin(3x)

The derivative of

\cos(u)

-\sin(u) \cdot u'

Differentiate the exponential term

e^x

\frac{d}{dx}(e^x) = e^x

The exponential function

e^x

is its own derivative

Combine the results using the sum rule

\frac{dy}{dx} = -3\sin(3x) + e^x

The derivative of a sum of functions is the sum of their individual derivatives

Differentiation of trigonometric and exponential functions using the chain rule

Question