Can you help me with getting the derivatives of the following two functions please. \begin{gather*} f_1(x)=3^{\sin x}5^{\cos x} \\ f_2(x)=e^{x^2}+\sin^2 x. \end{gather*}
It is too complicated for me. Could someone provide me with some direction. Thank you.
I’ll walk you through the second one and let you try the first one again on your own.
We have $f_2(c)=e^{x^2}+\sin^2x$. A sum can always be differentiated one term at a time, so let’s concentrate on differentiating $e^{x^2}$. The rule for differentiating an exponential is pretty easy:
$$\frac{d}{dx}e^u=e^u\frac{du}{dx}\;.$$
In other words, you get the same exponential back again, but multiplied by the derivative of the exponent. Here $u=x^2$, so $\frac{du}{dx}=2x$, and
$$\frac{d}{dx}e^{x^2}=e^{x^2}\cdot(2x)=2xe^{x^2}\;.$$
Now we’ll deal with $\sin^2x$. Remember that this notation really means $(\sin x)^2$, so we’re differentiating a power:
$$\frac{d}{dx}(\sin x)^2=(2\sin x)\cdot\frac{d}{dx}\sin x\;.$$
Since the derivative of $\sin x$ is $\cos x$, we can finish the differentiation to get
$$\frac{d}{dx}(\sin x)^2=(2\sin x)(\cos x)=2\sin x\cos x\;.$$
Now just add the pieces:
$$f_2'(x)=2xe^{x^2}+2\sin x\cos x\;.$$
Differentiations may sometimes be long, but they’re never really complicated: you can always do them one step at a time, following pretty mechanical rules.
